Viết chương trình hiển thị tam giác pascal

Tam giác Pascal là một trong những ví dụ kinh điển được dạy cho sinh viên kỹ thuật. Nó có nhiều cách hiểu. Một trong những cách nổi tiếng là việc sử dụng nó với các phương trình nhị thức

Tất cả các giá trị bên ngoài tam giác được coi là không (0). Hàng đầu tiên là 0 1 0 trong khi chỉ có 1 có khoảng trắng trong tam giác pascal, 0 là vô hình. Hàng thứ hai có được bằng cách cộng (0+1) và (1+0). Đầu ra được kẹp giữa hai số không. Quá trình tiếp tục cho đến khi đạt được mức yêu cầu

Tam giác Pascal có thể được suy ra bằng định lý nhị thức. Chúng ta có thể sử dụng các kết hợp và giai thừa để đạt được điều này

thuật toán

Giả sử rằng chúng ta đã biết rõ về giai thừa, chúng ta sẽ xem xét khái niệm cốt lõi của việc vẽ một tam giác pascal theo kiểu từng bước -

START
  Step  1 - Take number of rows to be printed, n.
  Step  2 - Make outer iteration I for n times to print rows
  Step  3 - Make inner iteration for J to (N - 1)
  Step  4 - Print single blank space " "
  Step  5 - Close inner loop
  Step  6 - Make inner iteration for J to I
  Step  7 - Print nCr of I and J
  Step  8 - Close inner loop
  Step  9 - Print NEWLINE character after each inner iteration
  Step 10 - Return
STOP

mã giả

Chúng ta có thể rút ra một mã giả cho thuật toán đã đề cập ở trên, như sau -

procedure pascals_triangle

   FOR I = 0 to N DO
      FOR J = 0 to N-1 DO
         PRINT " "
      END FOR

      FOR J = 0 to I DO
         PRINT nCr(i,j)
      END FOR

      PRINT NEWLINE
   END FOR

end procedure

Thực hiện

Hãy thực hiện toàn bộ chương trình này. Chúng ta sẽ triển khai các hàm cho giai thừa (không đệ quy) cũng như ncr (kết hợp)

Vòng lặp bên trong đầu tiên hiển thị không gian trên màn hình đầu ra. nếu bạn không muốn hiển thị đầu ra ở giữa màn hình thì hãy xóa cái này để lặp lại

Trong tam giác pascal, ở mỗi hàng, số đầu tiên và số cuối cùng là 1 và các số còn lại là tổng của hai số ngay phía trên nó. Vì vậy, bên trong vòng lặp bên trong thứ hai, chúng tôi viết điều kiện dưới đây,

if(j==0||j==i) pascal[i][j]=1;
else pascal[i][j] = pascal[i-1][j-1] + pascal[i-1][j];

Lưu ý rằng trong mỗi hàng, kích thước của mảng là n, nhưng ở hàng đầu tiên, phần tử đầu tiên duy nhất được lấp đầy và phần còn lại có giá trị rác. Tương tự, ở hàng thứ hai, chỉ phần tử thứ nhất và thứ hai của mảng được lấp đầy và phần còn lại có giá trị rác. Chúng tôi không muốn hiển thị giá trị rác. Vì vậy, chúng ta sẽ dễ dàng hiển thị đầu ra tại thời điểm tính toán

Tìm hiểu thêm. - Những bí mật toán học của tam giác Pascal

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Tam giác Pascal là một mảng tam giác các hệ số nhị thức. Viết hàm nhập vào một giá trị nguyên n và in ra n dòng đầu tiên của tam giác Pascal. Sau đây là 6 hàng đầu tiên của Tam giác Pascal

1  
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 
1 5 10 10 5 1 

Đề nghị thực hành

Tam giác Pascal

Thử nó

Phương pháp 1 ( Độ phức tạp thời gian O(n^3) )
Số lượng mục trong mỗi dòng bằng số dòng. Ví dụ, dòng đầu tiên có “1”, dòng thứ hai có “1 1”, dòng thứ ba có “1 2 1”,. và như thế. Mỗi mục trong một dòng là giá trị của Hệ số nhị thức. Giá trị của mục thứ i trong dòng số dòng là C(line, i). Giá trị có thể được tính bằng công thức sau.  

C(line, i)   = line! / ( (line-i)! * i! ) 

Một phương pháp đơn giản là chạy hai vòng lặp và tính giá trị của Hệ số nhị thức trong vòng lặp bên trong.  

C++

//  C++ code for Pascal's Triangle

#include

using namespace

C(line, i)   = line! / ( (line-i)! * i! ) 

0

C(line, i)   = line! / ( (line-i)! * i! ) 

1

C(line, i)   = line! / ( (line-i)! * i! ) 

2

C(line, i)   = line! / ( (line-i)! * i! ) 

3

C(line, i)   = line! / ( (line-i)! * i! ) 

4

C(line, i)   = line! / ( (line-i)! * i! ) 

5_______4_______4

C(line, i)   = line! / ( (line-i)! * i! ) 

7

C(line, i)   = line! / ( (line-i)! * i! ) 

4

C(line, i)   = line! / ( (line-i)! * i! ) 

9

C(line, i)   = line! / ( (line-i)! * i! ) 

1

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

1

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

2

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

3

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

4

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

5_______4_______4

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

7

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

0

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

2

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5

C(line, i)   = line! / ( (line-i)! * i! ) 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

7

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

1

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

3

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

5

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5

C(line, i)   = line! / ( (line-i)! * i! ) 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

0

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

1

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

2

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

3

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

2

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

7

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

8

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9//  C++ code for Pascal's Triangle0

//  C++ code for Pascal's Triangle0

C(line, i)   = line! / ( (line-i)! * i! ) 

1

//  C++ code for Pascal's Triangle3

C(line, i)   = line! / ( (line-i)! * i! ) 

3

C(line, i)   = line! / ( (line-i)! * i! ) 

4

C(line, i)   = line! / ( (line-i)! * i! ) 

5_______4_______4

C(line, i)   = line! / ( (line-i)! * i! ) 

7

C(line, i)   = line! / ( (line-i)! * i! ) 

4 #include 0

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

4 #include 4

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9#include 6 #include 7

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9#include 9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5

C(line, i)   = line! / ( (line-i)! * i! ) 

4 using4

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0using8

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0namespace0

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9//  C++ code for Pascal's Triangle0

namespace3

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9namespace5 namespace6

//  C++ code for Pascal's Triangle0

C(line, i)   = line! / ( (line-i)! * i! ) 

1

namespace9

C(line, i)   = line! / ( (line-i)! * i! ) 

4

C(line, i)   = line! / ( (line-i)! * i! ) 

01

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

4

C(line, i)   = line! / ( (line-i)! * i! ) 

05

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

07

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9namespace5

C(line, i)   = line! / ( (line-i)! * i! ) 

10

//  C++ code for Pascal's Triangle0

C(line, i)   = line! / ( (line-i)! * i! ) 

1

C(line, i)   = line! / ( (line-i)! * i! ) 

13

C

//  C++ code for Pascal's Triangle

C(line, i)   = line! / ( (line-i)! * i! ) 

15

C(line, i)   = line! / ( (line-i)! * i! ) 

1

C(line, i)   = line! / ( (line-i)! * i! ) 

2

C(line, i)   = line! / ( (line-i)! * i! ) 

3

C(line, i)   = line! / ( (line-i)! * i! ) 

4

C(line, i)   = line! / ( (line-i)! * i! ) 

5_______4_______4

C(line, i)   = line! / ( (line-i)! * i! ) 

7

C(line, i)   = line! / ( (line-i)! * i! ) 

4

C(line, i)   = line! / ( (line-i)! * i! ) 

9

C(line, i)   = line! / ( (line-i)! * i! ) 

1

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

1

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

2

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

3

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

4

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

5_______4_______4

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

7

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

0

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

2

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5

C(line, i)   = line! / ( (line-i)! * i! ) 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

7

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

1

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

3

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

5

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5

C(line, i)   = line! / ( (line-i)! * i! ) 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

0

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

1

C(line, i)   = line! / ( (line-i)! * i! ) 

57

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5

C(line, i)   = line! / ( (line-i)! * i! ) 

59

C(line, i)   = line! / ( (line-i)! * i! ) 

60

C(line, i)   = line! / ( (line-i)! * i! ) 

61_______4_______62

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

C(line, i)   = line! / ( (line-i)! * i! ) 

57

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

7

C(line, i)   = line! / ( (line-i)! * i! ) 

67

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9//  C++ code for Pascal's Triangle0

//  C++ code for Pascal's Triangle0

C(line, i)   = line! / ( (line-i)! * i! ) 

1

//  C++ code for Pascal's Triangle3

C(line, i)   = line! / ( (line-i)! * i! ) 

3

C(line, i)   = line! / ( (line-i)! * i! ) 

4

C(line, i)   = line! / ( (line-i)! * i! ) 

5_______4_______4

C(line, i)   = line! / ( (line-i)! * i! ) 

7

C(line, i)   = line! / ( (line-i)! * i! ) 

4 #include 0

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

4 #include 4

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9#include 6 #include 7

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9#include 9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5

C(line, i)   = line! / ( (line-i)! * i! ) 

4 using4

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0using8

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0namespace0

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9//  C++ code for Pascal's Triangle0

namespace3

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9namespace5 namespace6

//  C++ code for Pascal's Triangle0

C(line, i)   = line! / ( (line-i)! * i! ) 

1

namespace9

C(line, i)   = line! / ( (line-i)! * i! ) 

4

C(line, i)   = line! / ( (line-i)! * i! ) 

01

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

4

C(line, i)   = line! / ( (line-i)! * i! ) 

05

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

07

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9namespace5

C(line, i)   = line! / ( (line-i)! * i! ) 

10

//  C++ code for Pascal's Triangle0

Java

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

21

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

22

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

23

C(line, i)   = line! / ( (line-i)! * i! ) 

1

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

25

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

26

namespace3

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

1

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

31

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

33

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

4

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

5

C(line, i)   = line! / ( (line-i)! * i! ) 

4

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

7

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

40

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

42

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

44

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5

C(line, i)   = line! / ( (line-i)! * i! ) 

4

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

49

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

50

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

51

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

1

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

57

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5

C(line, i)   = line! / ( (line-i)! * i! ) 

4

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

62

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

50

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

64

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

66

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

67

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

68

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

3

C(line, i)   = line! / ( (line-i)! * i! ) 

67

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

71

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

73

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9//  C++ code for Pascal's Triangle0

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9//  C++ code for Pascal's Triangle0

namespace3

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

80

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

82

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

33

C(line, i)   = line! / ( (line-i)! * i! ) 

4

C(line, i)   = line! / ( (line-i)! * i! ) 

5_______4_______4

C(line, i)   = line! / ( (line-i)! * i! ) 

7

C(line, i)   = line! / ( (line-i)! * i! ) 

4 #include 0

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

C(line, i)   = line! / ( (line-i)! * i! ) 

4

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

95

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

96

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

8

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

40

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0#include 6 #include 7

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0#include 9

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

04

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5

C(line, i)   = line! / ( (line-i)! * i! ) 

4

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

62

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

50

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

11

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

1using8

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

1

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

17

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

96

C(line, i)   = line! / ( (line-i)! * i! ) 

67

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0//  C++ code for Pascal's Triangle0

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0namespace5 namespace6

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9//  C++ code for Pascal's Triangle0

namespace3

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

29

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

31

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

33

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

34

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

39

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

40

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

8

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

07

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9//  C++ code for Pascal's Triangle0

//  C++ code for Pascal's Triangle0

C(line, i)   = line! / ( (line-i)! * i! ) 

1

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

48

Python3

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

49

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

50

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

51

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

52

C(line, i)   = line! / ( (line-i)! * i! ) 

1

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

54

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

55

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

52

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

57

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

58

namespace3

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

61

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

63

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

66

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

67

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

68

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

50

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

71

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

40

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

74

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

76

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

78

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

81

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

67

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

68

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

50

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

86

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

87

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

96

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

89

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

1_______25_______91

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

92

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

93

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

3

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

95

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

96

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

97

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

91

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

00

namespace3

C(line, i)   = line! / ( (line-i)! * i! ) 

1

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

03

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

04

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

57

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

06

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

08

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

96

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

96

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9#include 6 ________35 _______13________35 _______14 ________35 ______15

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

17_______25_______96

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

19

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

14

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

17

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

81_______25_______67

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

68

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

50

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

29

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

08

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

96

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

08

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

34

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

35

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

14

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

37

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

08

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

96

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

08

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

42

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

42

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

44

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

87

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

96

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

47

namespace3

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9namespace5

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

08

C(line, i)   = line! / ( (line-i)! * i! ) 

1

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

53

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

19

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

96

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

40

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

57

C(line, i)   = line! / ( (line-i)! * i! ) 

1

C(line, i)   = line! / ( (line-i)! * i! ) 

1

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

60

C#

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

61

using

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

63

C(line, i)   = line! / ( (line-i)! * i! ) 

1

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

25

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

26

namespace3

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

1

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

31

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

33

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

4

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

5

C(line, i)   = line! / ( (line-i)! * i! ) 

4

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

7

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

40

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

42

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

44

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5

C(line, i)   = line! / ( (line-i)! * i! ) 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

7

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

1

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

57

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5

C(line, i)   = line! / ( (line-i)! * i! ) 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

0

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

02

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

67

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

68

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

3

C(line, i)   = line! / ( (line-i)! * i! ) 

67

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

71

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

09

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9//  C++ code for Pascal's Triangle0

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9//  C++ code for Pascal's Triangle0

namespace3

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

80

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

82

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

33

C(line, i)   = line! / ( (line-i)! * i! ) 

4

C(line, i)   = line! / ( (line-i)! * i! ) 

5_______4_______4

C(line, i)   = line! / ( (line-i)! * i! ) 

7

C(line, i)   = line! / ( (line-i)! * i! ) 

4 #include 0

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

C(line, i)   = line! / ( (line-i)! * i! ) 

4 #include 4

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

40

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0#include 6 #include 7

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0#include 9

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

04

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5

C(line, i)   = line! / ( (line-i)! * i! ) 

4 using4

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

1using8

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

1namespace0

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0//  C++ code for Pascal's Triangle0

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0namespace5 namespace6

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9//  C++ code for Pascal's Triangle0

namespace3

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

29

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

31

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

33

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

64

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

4

C(line, i)   = line! / ( (line-i)! * i! ) 

05

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

07

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9//  C++ code for Pascal's Triangle0

//  C++ code for Pascal's Triangle0

C(line, i)   = line! / ( (line-i)! * i! ) 

1

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

76

PHP

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

77

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

78

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

79

C(line, i)   = line! / ( (line-i)! * i! ) 

1

C(line, i)   = line! / ( (line-i)! * i! ) 

3

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

82

C(line, i)   = line! / ( (line-i)! * i! ) 

5_______25_______84

C(line, i)   = line! / ( (line-i)! * i! ) 

60

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

86

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

47

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

90

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

91

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9#include 6

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

86

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

96

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

84

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

14

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

86

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

47

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

86

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

96

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

84

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

14

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

86

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

8

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5//  C++ code for Pascal's Triangle11 //  C++ code for Pascal's Triangle12//  C++ code for Pascal's Triangle11 //  C++ code for Pascal's Triangle14

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

86//  C++ code for Pascal's Triangle16//  C++ code for Pascal's Triangle11

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

47

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

90 //  C++ code for Pascal's Triangle23

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

84

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

14//  C++ code for Pascal's Triangle11

C(line, i)   = line! / ( (line-i)! * i! ) 

67

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

90 //  C++ code for Pascal's Triangle30//  C++ code for Pascal's Triangle11 //  C++ code for Pascal's Triangle32

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9//  C++ code for Pascal's Triangle0

namespace5

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

90

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

8

//  C++ code for Pascal's Triangle0

C(line, i)   = line! / ( (line-i)! * i! ) 

1

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

1

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

2

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

3

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

82

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

5

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

84

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

47

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

0

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

2

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5//  C++ code for Pascal's Triangle55 //  C++ code for Pascal's Triangle12//  C++ code for Pascal's Triangle55 //  C++ code for Pascal's Triangle14

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

84

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

8//  C++ code for Pascal's Triangle55//  C++ code for Pascal's Triangle62

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

1

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

3

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

5

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5//  C++ code for Pascal's Triangle11 //  C++ code for Pascal's Triangle12//  C++ code for Pascal's Triangle11 //  C++ code for Pascal's Triangle77//  C++ code for Pascal's Triangle55

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

8//  C++ code for Pascal's Triangle11//  C++ code for Pascal's Triangle62

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

93//  C++ code for Pascal's Triangle83 //  C++ code for Pascal's Triangle84//  C++ code for Pascal's Triangle85//  C++ code for Pascal's Triangle55

C(line, i)   = line! / ( (line-i)! * i! ) 

60//  C++ code for Pascal's Triangle11//  C++ code for Pascal's Triangle89

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

3

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

8

//  C++ code for Pascal's Triangle92

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0//  C++ code for Pascal's Triangle83

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

7

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

8

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9//  C++ code for Pascal's Triangle0

//  C++ code for Pascal's Triangle0

C(line, i)   = line! / ( (line-i)! * i! ) 

1

#include 01

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

84#include 03

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

5

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

84

C(line, i)   = line! / ( (line-i)! * i! ) 

67

C(line, i)   = line! / ( (line-i)! * i! ) 

1

#include 08

#include 09

Javascript

#include 10

C(line, i)   = line! / ( (line-i)! * i! ) 

1

#include 12

C(line, i)   = line! / ( (line-i)! * i! ) 

1

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

1

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

31

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

82

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

57

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

40

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

42

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

44

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4 #include 30

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

1

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

57

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4 #include 39

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0#include 41

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

67

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

68

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

3

C(line, i)   = line! / ( (line-i)! * i! ) 

67

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

71

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0#include 48#include 49_______4_______67

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9//  C++ code for Pascal's Triangle0

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9//  C++ code for Pascal's Triangle0

namespace3

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

80

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

82

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

82 #include 62

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0#include 66

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

40

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0#include 6 #include 7

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0#include 9

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

04

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4 #include 76

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

1using8

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

1namespace0

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0//  C++ code for Pascal's Triangle0

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0namespace5 namespace6

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9//  C++ code for Pascal's Triangle0

C(line, i)   = line! / ( (line-i)! * i! ) 

1

C(line, i)   = line! / ( (line-i)! * i! ) 

1

#include 01

C(line, i)   = line! / ( (line-i)! * i! ) 

1

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9#include 95

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

07

C(line, i)   = line! / ( (line-i)! * i! ) 

1

#include 99

Đầu ra

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

Độ phức tạp về thời gian. O(n^3)
Không gian phụ trợ. Ô(1)

Phương pháp 2( O(n^2) thời gian và O(n^2) không gian thêm)
Nếu chúng ta nhìn kỹ hơn vào tam giác, chúng ta sẽ quan sát thấy rằng mọi mục nhập đều là tổng của hai giá trị phía trên nó. Vì vậy, chúng ta có thể tạo một mảng 2D lưu trữ các giá trị được tạo trước đó. Để tạo một giá trị trong một dòng, chúng ta có thể sử dụng các giá trị được lưu trữ trước đó từ mảng.  
 

 

C++

using00

using01

using02

using03

using namespace

C(line, i)   = line! / ( (line-i)! * i! ) 

0

C(line, i)   = line! / ( (line-i)! * i! ) 

1

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

4

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

5_______4_______4

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

7

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

namespace3

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9using15

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9using17

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

4 using20

C(line, i)   = line! / ( (line-i)! * i! ) 

1

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9using23

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9using25

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5

C(line, i)   = line! / ( (line-i)! * i! ) 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

7

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0using34

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0using36

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5

C(line, i)   = line! / ( (line-i)! * i! ) 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

0

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0using45

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0#include 6 using48

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

1____1562_______50

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0using52

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0using54

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0using56

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

1using58

using59_______1562_______60

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0using62

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

3

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

8

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0//  C++ code for Pascal's Triangle0

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

2

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

7

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

8

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9//  C++ code for Pascal's Triangle0

//  C++ code for Pascal's Triangle0

C(line, i)   = line! / ( (line-i)! * i! ) 

1

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

29

C(line, i)   = line! / ( (line-i)! * i! ) 

4

C(line, i)   = line! / ( (line-i)! * i! ) 

01

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

4 using81

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

07

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9namespace5

C(line, i)   = line! / ( (line-i)! * i! ) 

10

//  C++ code for Pascal's Triangle0

C(line, i)   = line! / ( (line-i)! * i! ) 

1

using89

C

using90

using01

using02

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

4

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

5_______4_______4

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

7

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

using15

using17

C(line, i)   = line! / ( (line-i)! * i! ) 

4 using20

C(line, i)   = line! / ( (line-i)! * i! ) 

1

namespace03

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5

C(line, i)   = line! / ( (line-i)! * i! ) 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

7

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9using34

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9using36

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5

C(line, i)   = line! / ( (line-i)! * i! ) 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

0

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9using45

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9#include 6 using48

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0using50

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9using52

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9using54

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9using56

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0namespace34

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

57

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5

C(line, i)   = line! / ( (line-i)! * i! ) 

59namespace39

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9//  C++ code for Pascal's Triangle0

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

57

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

7

C(line, i)   = line! / ( (line-i)! * i! ) 

67

//  C++ code for Pascal's Triangle0

//  C++ code for Pascal's Triangle0

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

29

C(line, i)   = line! / ( (line-i)! * i! ) 

4

C(line, i)   = line! / ( (line-i)! * i! ) 

01

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

C(line, i)   = line! / ( (line-i)! * i! ) 

4 using81

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

07

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9namespace5

C(line, i)   = line! / ( (line-i)! * i! ) 

10

//  C++ code for Pascal's Triangle0

Java

namespace61

namespace62

namespace63

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

22

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

23

C(line, i)   = line! / ( (line-i)! * i! ) 

1

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

25

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

26

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

31

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

33

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

4 namespace73

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

C(line, i)   = line! / ( (line-i)! * i! ) 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

39namespace77

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

8

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

C(line, i)   = line! / ( (line-i)! * i! ) 

07

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9//  C++ code for Pascal's Triangle0

C(line, i)   = line! / ( (line-i)! * i! ) 

1

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

31

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

33

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

4

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

5_______4_______4

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

7

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

namespace91

C(line, i)   = line! / ( (line-i)! * i! ) 

4namespace93namespace94

C(line, i)   = line! / ( (line-i)! * i! ) 

4namespace96

C(line, i)   = line! / ( (line-i)! * i! ) 

1

namespace03

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5

C(line, i)   = line! / ( (line-i)! * i! ) 

4

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

49

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

50

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

51

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

007

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5

C(line, i)   = line! / ( (line-i)! * i! ) 

4

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

62

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

50

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

64

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9using45

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9#include 6

C(line, i)   = line! / ( (line-i)! * i! ) 

021

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

50

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

47

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

C(line, i)   = line! / ( (line-i)! * i! ) 

025

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

96

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

8

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9using56

C(line, i)   = line! / ( (line-i)! * i! ) 

030

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

C(line, i)   = line! / ( (line-i)! * i! ) 

032

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

96

C(line, i)   = line! / ( (line-i)! * i! ) 

034

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

96

C(line, i)   = line! / ( (line-i)! * i! ) 

036

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

96

C(line, i)   = line! / ( (line-i)! * i! ) 

038

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

040

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9//  C++ code for Pascal's Triangle0

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

044//  C++ code for Pascal's Triangle84

C(line, i)   = line! / ( (line-i)! * i! ) 

67

//  C++ code for Pascal's Triangle0

//  C++ code for Pascal's Triangle0

//  C++ code for Pascal's Triangle0

Python3

C(line, i)   = line! / ( (line-i)! * i! ) 

050

C(line, i)   = line! / ( (line-i)! * i! ) 

1

C(line, i)   = line! / ( (line-i)! * i! ) 

052

C(line, i)   = line! / ( (line-i)! * i! ) 

053

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

57

C(line, i)   = line! / ( (line-i)! * i! ) 

055

C(line, i)   = line! / ( (line-i)! * i! ) 

4

C(line, i)   = line! / ( (line-i)! * i! ) 

057

C(line, i)   = line! / ( (line-i)! * i! ) 

1

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

060

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

062

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

064

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

96

C(line, i)   = line! / ( (line-i)! * i! ) 

066

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

50

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4

C(line, i)   = line! / ( (line-i)! * i! ) 

069

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

67

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

68

C(line, i)   = line! / ( (line-i)! * i! ) 

072

C(line, i)   = line! / ( (line-i)! * i! ) 

073

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4

C(line, i)   = line! / ( (line-i)! * i! ) 

075

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

67

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

68

C(line, i)   = line! / ( (line-i)! * i! ) 

072

C(line, i)   = line! / ( (line-i)! * i! ) 

1

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9____4_______081

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9____4_______083

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

66

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

67

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

68

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

50

C(line, i)   = line! / ( (line-i)! * i! ) 

091

C(line, i)   = line! / ( (line-i)! * i! ) 

1

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

74

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

C(line, i)   = line! / ( (line-i)! * i! ) 

096

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

81

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

67

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

68

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

50

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

86

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

87

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

96

C(line, i)   = line! / ( (line-i)! * i! ) 

057

C(line, i)   = line! / ( (line-i)! * i! ) 

1

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

1____4_______110

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

1____4_______112

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

1#include 6

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

44_______4_______116

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

50

C(line, i)   = line! / ( (line-i)! * i! ) 

118

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

81

C(line, i)   = line! / ( (line-i)! * i! ) 

116

C(line, i)   = line! / ( (line-i)! * i! ) 

121

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

93_______4_______123____25_______96

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

96

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

93

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

91

C(line, i)   = line! / ( (line-i)! * i! ) 

128

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

96

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

3

C(line, i)   = line! / ( (line-i)! * i! ) 

131

C(line, i)   = line! / ( (line-i)! * i! ) 

1

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

1____4_______134

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

1

C(line, i)   = line! / ( (line-i)! * i! ) 

136

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

1____1562_______56

C(line, i)   = line! / ( (line-i)! * i! ) 

139

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

93

C(line, i)   = line! / ( (line-i)! * i! ) 

123

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

96

C(line, i)   = line! / ( (line-i)! * i! ) 

143

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

14

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

96

C(line, i)   = line! / ( (line-i)! * i! ) 

146

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

14

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

96

C(line, i)   = line! / ( (line-i)! * i! ) 

149

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

87

C(line, i)   = line! / ( (line-i)! * i! ) 

151

C(line, i)   = line! / ( (line-i)! * i! ) 

152

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

14

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

96

C(line, i)   = line! / ( (line-i)! * i! ) 

155

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

93

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

91

C(line, i)   = line! / ( (line-i)! * i! ) 

128

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

96

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

3

C(line, i)   = line! / ( (line-i)! * i! ) 

161

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

91

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

7

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

95

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

96

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

97

C(line, i)   = line! / ( (line-i)! * i! ) 

1

C(line, i)   = line! / ( (line-i)! * i! ) 

170

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

19

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

96 namespace77

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

57

C(line, i)   = line! / ( (line-i)! * i! ) 

1

C(line, i)   = line! / ( (line-i)! * i! ) 

176

C(line, i)   = line! / ( (line-i)! * i! ) 

177

C#

C(line, i)   = line! / ( (line-i)! * i! ) 

178

namespace62

namespace63

using

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

63

C(line, i)   = line! / ( (line-i)! * i! ) 

1

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

25

C(line, i)   = line! / ( (line-i)! * i! ) 

185

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

31

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

33

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

4

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

5_______4_______4

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

7

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

namespace3

using15

using17

C(line, i)   = line! / ( (line-i)! * i! ) 

4

C(line, i)   = line! / ( (line-i)! * i! ) 

198namespace94

C(line, i)   = line! / ( (line-i)! * i! ) 

4

C(line, i)   = line! / ( (line-i)! * i! ) 

201

C(line, i)   = line! / ( (line-i)! * i! ) 

1

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

42

C(line, i)   = line! / ( (line-i)! * i! ) 

204

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5

C(line, i)   = line! / ( (line-i)! * i! ) 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

7

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

1

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

57

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5

C(line, i)   = line! / ( (line-i)! * i! ) 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

0

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

40

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

223

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

225

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9#include 6 using48

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

C(line, i)   = line! / ( (line-i)! * i! ) 

230

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9using56

C(line, i)   = line! / ( (line-i)! * i! ) 

233

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

40_______4_______235

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

C(line, i)   = line! / ( (line-i)! * i! ) 

237

C(line, i)   = line! / ( (line-i)! * i! ) 

238______4_______239

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

241

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9//  C++ code for Pascal's Triangle0

C(line, i)   = line! / ( (line-i)! * i! ) 

244____1560_______84

C(line, i)   = line! / ( (line-i)! * i! ) 

67

//  C++ code for Pascal's Triangle0

//  C++ code for Pascal's Triangle0

C(line, i)   = line! / ( (line-i)! * i! ) 

1

#include 01

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

31

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

33

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

4

C(line, i)   = line! / ( (line-i)! * i! ) 

254

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

4 using81

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

07

//  C++ code for Pascal's Triangle0

//  C++ code for Pascal's Triangle0

C(line, i)   = line! / ( (line-i)! * i! ) 

1

C(line, i)   = line! / ( (line-i)! * i! ) 

264

C(line, i)   = line! / ( (line-i)! * i! ) 

265

PHP

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

77

C(line, i)   = line! / ( (line-i)! * i! ) 

267

using01

using02

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

82

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

5

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

84

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

47

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9using15

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9using17

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

280

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

96

C(line, i)   = line! / ( (line-i)! * i! ) 

282

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5

C(line, i)   = line! / ( (line-i)! * i! ) 

282

C(line, i)   = line! / ( (line-i)! * i! ) 

285

namespace3

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9using23

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9using25

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5//  C++ code for Pascal's Triangle55 //  C++ code for Pascal's Triangle12//  C++ code for Pascal's Triangle55 //  C++ code for Pascal's Triangle14

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

84

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

8//  C++ code for Pascal's Triangle55//  C++ code for Pascal's Triangle62

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0using34

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0using36

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5//  C++ code for Pascal's Triangle11 //  C++ code for Pascal's Triangle12//  C++ code for Pascal's Triangle11 //  C++ code for Pascal's Triangle77//  C++ code for Pascal's Triangle55

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

8//  C++ code for Pascal's Triangle11//  C++ code for Pascal's Triangle62

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

1using45

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

1#include 6

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5//  C++ code for Pascal's Triangle55

C(line, i)   = line! / ( (line-i)! * i! ) 

327//  C++ code for Pascal's Triangle11

C(line, i)   = line! / ( (line-i)! * i! ) 

329//  C++ code for Pascal's Triangle11

C(line, i)   = line! / ( (line-i)! * i! ) 

331

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

93

C(line, i)   = line! / ( (line-i)! * i! ) 

280

C(line, i)   = line! / ( (line-i)! * i! ) 

334//  C++ code for Pascal's Triangle55

C(line, i)   = line! / ( (line-i)! * i! ) 

336//  C++ code for Pascal's Triangle11

C(line, i)   = line! / ( (line-i)! * i! ) 

338

//  C++ code for Pascal's Triangle92

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

1____1562_______52

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

1using54

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

1____1562_______56

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

93

C(line, i)   = line! / ( (line-i)! * i! ) 

280

C(line, i)   = line! / ( (line-i)! * i! ) 

334_______1560_______55

C(line, i)   = line! / ( (line-i)! * i! ) 

336//  C++ code for Pascal's Triangle11

C(line, i)   = line! / ( (line-i)! * i! ) 

352

C(line, i)   = line! / ( (line-i)! * i! ) 

280

C(line, i)   = line! / ( (line-i)! * i! ) 

334//  C++ code for Pascal's Triangle55

C(line, i)   = line! / ( (line-i)! * i! ) 

356//  C++ code for Pascal's Triangle11

C(line, i)   = line! / ( (line-i)! * i! ) 

358

C(line, i)   = line! / ( (line-i)! * i! ) 

151

C(line, i)   = line! / ( (line-i)! * i! ) 

280

C(line, i)   = line! / ( (line-i)! * i! ) 

334//  C++ code for Pascal's Triangle55

C(line, i)   = line! / ( (line-i)! * i! ) 

356//  C++ code for Pascal's Triangle11

C(line, i)   = line! / ( (line-i)! * i! ) 

365

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

1//  C++ code for Pascal's Triangle83

C(line, i)   = line! / ( (line-i)! * i! ) 

280

C(line, i)   = line! / ( (line-i)! * i! ) 

334//  C++ code for Pascal's Triangle55

C(line, i)   = line! / ( (line-i)! * i! ) 

336//  C++ code for Pascal's Triangle11

C(line, i)   = line! / ( (line-i)! * i! ) 

373

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

3

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

8

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0//  C++ code for Pascal's Triangle0

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0//  C++ code for Pascal's Triangle83

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

7

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

8

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9//  C++ code for Pascal's Triangle0

//  C++ code for Pascal's Triangle0

C(line, i)   = line! / ( (line-i)! * i! ) 

1

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

29

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

84

C(line, i)   = line! / ( (line-i)! * i! ) 

388

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

5

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

84

C(line, i)   = line! / ( (line-i)! * i! ) 

67

C(line, i)   = line! / ( (line-i)! * i! ) 

1

C(line, i)   = line! / ( (line-i)! * i! ) 

393

C(line, i)   = line! / ( (line-i)! * i! ) 

394

#include 09

Javascript

#include 10

C(line, i)   = line! / ( (line-i)! * i! ) 

1

C(line, i)   = line! / ( (line-i)! * i! ) 

398

namespace62

namespace63

C(line, i)   = line! / ( (line-i)! * i! ) 

401 using81

C(line, i)   = line! / ( (line-i)! * i! ) 

07

C(line, i)   = line! / ( (line-i)! * i! ) 

1

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

82

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

57

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

namespace91

C(line, i)   = line! / ( (line-i)! * i! ) 

409

C(line, i)   = line! / ( (line-i)! * i! ) 

1

namespace03

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4

C(line, i)   = line! / ( (line-i)! * i! ) 

413

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

007

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4

C(line, i)   = line! / ( (line-i)! * i! ) 

419

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9using45

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9#include 6 using48

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0using50

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9using56

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

030

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0namespace34

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

436

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9//  C++ code for Pascal's Triangle0

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9#include 48

C(line, i)   = line! / ( (line-i)! * i! ) 

441

C(line, i)   = line! / ( (line-i)! * i! ) 

67

//  C++ code for Pascal's Triangle0

//  C++ code for Pascal's Triangle0

C(line, i)   = line! / ( (line-i)! * i! ) 

1

C(line, i)   = line! / ( (line-i)! * i! ) 

446

C(line, i)   = line! / ( (line-i)! * i! ) 

1

#include 99

Đầu ra

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

Phương pháp này có thể được tối ưu hóa để sử dụng thêm không gian O(n) vì chúng tôi chỉ cần các giá trị từ hàng trước đó. Vì vậy, chúng ta có thể tạo một mảng phụ có kích thước n và ghi đè lên các giá trị. Sau đây là một phương pháp khác chỉ sử dụng không gian thừa O(1)
Phương pháp 3 ( O(n^2) thời gian và O(1) không gian thêm)
Phương pháp này dựa trên phương pháp 1. Chúng ta biết rằng mục thứ i trong một dòng số là Hệ số nhị thức C(dòng, i) và tất cả các dòng bắt đầu bằng giá trị 1. Ý tưởng là tính toán C(dòng, i) bằng cách sử dụng C(dòng, i-1). Nó có thể được tính trong thời gian O(1) bằng cách sử dụng như sau.  
 

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

C++

using00

C(line, i)   = line! / ( (line-i)! * i! ) 

450

C(line, i)   = line! / ( (line-i)! * i! ) 

451

using03

C(line, i)   = line! / ( (line-i)! * i! ) 

1

using namespace

C(line, i)   = line! / ( (line-i)! * i! ) 

0

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

4

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

5_______4_______4

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

7

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

namespace3

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5

C(line, i)   = line! / ( (line-i)! * i! ) 

4

C(line, i)   = line! / ( (line-i)! * i! ) 

466

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

4

C(line, i)   = line! / ( (line-i)! * i! ) 

470

C(line, i)   = line! / ( (line-i)! * i! ) 

471

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5

C(line, i)   = line! / ( (line-i)! * i! ) 

4

C(line, i)   = line! / ( (line-i)! * i! ) 

476

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

40

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

C(line, i)   = line! / ( (line-i)! * i! ) 

481

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

C(line, i)   = line! / ( (line-i)! * i! ) 

483

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

3

C(line, i)   = line! / ( (line-i)! * i! ) 

485

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

C(line, i)   = line! / ( (line-i)! * i! ) 

487

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9//  C++ code for Pascal's Triangle0

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

491

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

7

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

8

//  C++ code for Pascal's Triangle0

//  C++ code for Pascal's Triangle0

C(line, i)   = line! / ( (line-i)! * i! ) 

1

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

29

C(line, i)   = line! / ( (line-i)! * i! ) 

4

C(line, i)   = line! / ( (line-i)! * i! ) 

01

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

4 using81

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

07

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9namespace5

C(line, i)   = line! / ( (line-i)! * i! ) 

10

//  C++ code for Pascal's Triangle0

C(line, i)   = line! / ( (line-i)! * i! ) 

1

C(line, i)   = line! / ( (line-i)! * i! ) 

511

C

using90

C(line, i)   = line! / ( (line-i)! * i! ) 

450

C(line, i)   = line! / ( (line-i)! * i! ) 

451

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

4

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

5_______4_______4

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

7

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5

C(line, i)   = line! / ( (line-i)! * i! ) 

4

C(line, i)   = line! / ( (line-i)! * i! ) 

466

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

4

C(line, i)   = line! / ( (line-i)! * i! ) 

470

C(line, i)   = line! / ( (line-i)! * i! ) 

471

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5

C(line, i)   = line! / ( (line-i)! * i! ) 

4

C(line, i)   = line! / ( (line-i)! * i! ) 

476

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

57

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5

C(line, i)   = line! / ( (line-i)! * i! ) 

59

C(line, i)   = line! / ( (line-i)! * i! ) 

540

C(line, i)   = line! / ( (line-i)! * i! ) 

481

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

487

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9//  C++ code for Pascal's Triangle0

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

57

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

7

C(line, i)   = line! / ( (line-i)! * i! ) 

67

//  C++ code for Pascal's Triangle0

//  C++ code for Pascal's Triangle0

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

29

C(line, i)   = line! / ( (line-i)! * i! ) 

4

C(line, i)   = line! / ( (line-i)! * i! ) 

01

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

C(line, i)   = line! / ( (line-i)! * i! ) 

4 using81

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

07

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9namespace5

C(line, i)   = line! / ( (line-i)! * i! ) 

10

//  C++ code for Pascal's Triangle0

Java

C(line, i)   = line! / ( (line-i)! * i! ) 

565

C(line, i)   = line! / ( (line-i)! * i! ) 

566

namespace63

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

22

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

23

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

25

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

26

C(line, i)   = line! / ( (line-i)! * i! ) 

1

C(line, i)   = line! / ( (line-i)! * i! ) 

573

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

31

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

33

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

4

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

5_______4_______4

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

7

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5

C(line, i)   = line! / ( (line-i)! * i! ) 

4

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

49

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

96

C(line, i)   = line! / ( (line-i)! * i! ) 

587

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

40

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

4

C(line, i)   = line! / ( (line-i)! * i! ) 

593

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

96

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

8

C(line, i)   = line! / ( (line-i)! * i! ) 

471

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5

C(line, i)   = line! / ( (line-i)! * i! ) 

4

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

62

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

96

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

64

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

605

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

C(line, i)   = line! / ( (line-i)! * i! ) 

481

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

C(line, i)   = line! / ( (line-i)! * i! ) 

609

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

3

C(line, i)   = line! / ( (line-i)! * i! ) 

67

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

C(line, i)   = line! / ( (line-i)! * i! ) 

487

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9//  C++ code for Pascal's Triangle0

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

73

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9//  C++ code for Pascal's Triangle0

//  C++ code for Pascal's Triangle0

C(line, i)   = line! / ( (line-i)! * i! ) 

1

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

29

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

31

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

33

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

4 namespace73

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

39namespace77

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

8

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

07

C(line, i)   = line! / ( (line-i)! * i! ) 

634

//  C++ code for Pascal's Triangle0

C(line, i)   = line! / ( (line-i)! * i! ) 

264

C(line, i)   = line! / ( (line-i)! * i! ) 

637

Python3

C(line, i)   = line! / ( (line-i)! * i! ) 

638

C(line, i)   = line! / ( (line-i)! * i! ) 

639

C(line, i)   = line! / ( (line-i)! * i! ) 

640

C(line, i)   = line! / ( (line-i)! * i! ) 

1

C(line, i)   = line! / ( (line-i)! * i! ) 

642

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

57

C(line, i)   = line! / ( (line-i)! * i! ) 

644

C(line, i)   = line! / ( (line-i)! * i! ) 

1

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

66

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

67

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

68

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

96

C(line, i)   = line! / ( (line-i)! * i! ) 

653

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

87

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

96

C(line, i)   = line! / ( (line-i)! * i! ) 

656

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

C(line, i)   = line! / ( (line-i)! * i! ) 

658

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

96

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

96

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

8

C(line, i)   = line! / ( (line-i)! * i! ) 

662

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

81

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

67

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

68

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

96

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

86

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

87

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

96

C(line, i)   = line! / ( (line-i)! * i! ) 

656

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

04

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

1____4_______676

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

1____4_______678

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

1

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

91

C(line, i)   = line! / ( (line-i)! * i! ) 

681

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

96

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

3

C(line, i)   = line! / ( (line-i)! * i! ) 

684

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

1

C(line, i)   = line! / ( (line-i)! * i! ) 

658

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

96

C(line, i)   = line! / ( (line-i)! * i! ) 

4

C(line, i)   = line! / ( (line-i)! * i! ) 

689

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

34

C(line, i)   = line! / ( (line-i)! * i! ) 

691

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

14

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

37

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

42

C(line, i)   = line! / ( (line-i)! * i! ) 

695

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

91

C(line, i)   = line! / ( (line-i)! * i! ) 

698

C(line, i)   = line! / ( (line-i)! * i! ) 

1

C(line, i)   = line! / ( (line-i)! * i! ) 

700

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

19

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

96 namespace77

C(line, i)   = line! / ( (line-i)! * i! ) 

485

C(line, i)   = line! / ( (line-i)! * i! ) 

07

C(line, i)   = line! / ( (line-i)! * i! ) 

1

C(line, i)   = line! / ( (line-i)! * i! ) 

707

C#

C(line, i)   = line! / ( (line-i)! * i! ) 

708

C(line, i)   = line! / ( (line-i)! * i! ) 

566

C(line, i)   = line! / ( (line-i)! * i! ) 

710

using

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

63

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

25

C(line, i)   = line! / ( (line-i)! * i! ) 

185

C(line, i)   = line! / ( (line-i)! * i! ) 

605

C(line, i)   = line! / ( (line-i)! * i! ) 

1

C(line, i)   = line! / ( (line-i)! * i! ) 

717

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

31

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

33

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

4

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

5_______4_______4

C(line, i)   = line! / ( (line-i)! * i! ) 

723

C(line, i)   = line! / ( (line-i)! * i! ) 

605

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5

C(line, i)   = line! / ( (line-i)! * i! ) 

4

C(line, i)   = line! / ( (line-i)! * i! ) 

729

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

1_______4_______731

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

605

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

40

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

4

C(line, i)   = line! / ( (line-i)! * i! ) 

470

C(line, i)   = line! / ( (line-i)! * i! ) 

738

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5

C(line, i)   = line! / ( (line-i)! * i! ) 

4

C(line, i)   = line! / ( (line-i)! * i! ) 

476

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

605

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

C(line, i)   = line! / ( (line-i)! * i! ) 

747

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

C(line, i)   = line! / ( (line-i)! * i! ) 

749

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

C(line, i)   = line! / ( (line-i)! * i! ) 

751

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

3

C(line, i)   = line! / ( (line-i)! * i! ) 

684

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

C(line, i)   = line! / ( (line-i)! * i! ) 

487

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

634

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

759

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

7

C(line, i)   = line! / ( (line-i)! * i! ) 

761

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

634

C(line, i)   = line! / ( (line-i)! * i! ) 

634

C(line, i)   = line! / ( (line-i)! * i! ) 

1

C(line, i)   = line! / ( (line-i)! * i! ) 

766

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

31

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

33

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

4

C(line, i)   = line! / ( (line-i)! * i! ) 

770

C(line, i)   = line! / ( (line-i)! * i! ) 

605

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

4

C(line, i)   = line! / ( (line-i)! * i! ) 

774

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

776

C(line, i)   = line! / ( (line-i)! * i! ) 

634

C(line, i)   = line! / ( (line-i)! * i! ) 

634

C(line, i)   = line! / ( (line-i)! * i! ) 

1

C(line, i)   = line! / ( (line-i)! * i! ) 

393

C(line, i)   = line! / ( (line-i)! * i! ) 

781

PHP

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

77

C(line, i)   = line! / ( (line-i)! * i! ) 

783

C(line, i)   = line! / ( (line-i)! * i! ) 

566

C(line, i)   = line! / ( (line-i)! * i! ) 

710

C(line, i)   = line! / ( (line-i)! * i! ) 

1

C(line, i)   = line! / ( (line-i)! * i! ) 

717

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

82

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

5

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

84

C(line, i)   = line! / ( (line-i)! * i! ) 

131

C(line, i)   = line! / ( (line-i)! * i! ) 

605

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5//  C++ code for Pascal's Triangle55

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

91//  C++ code for Pascal's Triangle55 //  C++ code for Pascal's Triangle77

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

84

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

8//  C++ code for Pascal's Triangle55

C(line, i)   = line! / ( (line-i)! * i! ) 

803

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

605

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

C(line, i)   = line! / ( (line-i)! * i! ) 

807

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

91

C(line, i)   = line! / ( (line-i)! * i! ) 

738

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5//  C++ code for Pascal's Triangle11

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

91//  C++ code for Pascal's Triangle11 //  C++ code for Pascal's Triangle77//  C++ code for Pascal's Triangle55

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

8//  C++ code for Pascal's Triangle11

C(line, i)   = line! / ( (line-i)! * i! ) 

803

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

C(line, i)   = line! / ( (line-i)! * i! ) 

605

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

1_______4_______824

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

1_______4_______749

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

1

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

91

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5

C(line, i)   = line! / ( (line-i)! * i! ) 

807

C(line, i)   = line! / ( (line-i)! * i! ) 

831

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

3_______4_______684

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

1

C(line, i)   = line! / ( (line-i)! * i! ) 

807

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

96

C(line, i)   = line! / ( (line-i)! * i! ) 

807

C(line, i)   = line! / ( (line-i)! * i! ) 

838//  C++ code for Pascal's Triangle55

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

14//  C++ code for Pascal's Triangle11

C(line, i)   = line! / ( (line-i)! * i! ) 

842//  C++ code for Pascal's Triangle11

C(line, i)   = line! / ( (line-i)! * i! ) 

485

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

C(line, i)   = line! / ( (line-i)! * i! ) 

634

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

91

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

5

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

7

C(line, i)   = line! / ( (line-i)! * i! ) 

684

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

634

C(line, i)   = line! / ( (line-i)! * i! ) 

634

C(line, i)   = line! / ( (line-i)! * i! ) 

1

C(line, i)   = line! / ( (line-i)! * i! ) 

766

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

84

C(line, i)   = line! / ( (line-i)! * i! ) 

858

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

5

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

84

C(line, i)   = line! / ( (line-i)! * i! ) 

67

C(line, i)   = line! / ( (line-i)! * i! ) 

1

C(line, i)   = line! / ( (line-i)! * i! ) 

863

#include 09

Javascript

#include 10

C(line, i)   = line! / ( (line-i)! * i! ) 

1

C(line, i)   = line! / ( (line-i)! * i! ) 

867

C(line, i)   = line! / ( (line-i)! * i! ) 

566

namespace63

C(line, i)   = line! / ( (line-i)! * i! ) 

1

C(line, i)   = line! / ( (line-i)! * i! ) 

573

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

82

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

57

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4

C(line, i)   = line! / ( (line-i)! * i! ) 

877

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

8

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

40

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

401

C(line, i)   = line! / ( (line-i)! * i! ) 

883

C(line, i)   = line! / ( (line-i)! * i! ) 

471

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

4

C(line, i)   = line! / ( (line-i)! * i! ) 

887

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9

C(line, i)   = line! / ( (line-i)! * i! ) 

605

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

C(line, i)   = line! / ( (line-i)! * i! ) 

481

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

C(line, i)   = line! / ( (line-i)! * i! ) 

893

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

3

C(line, i)   = line! / ( (line-i)! * i! ) 

67

C(line, i)   = line! / ( (line-i)! * i! )
C(line, i-1) = line! / ( (line - i + 1)! * (i-1)! )
We can derive following expression from above two expressions.
C(line, i) = C(line, i-1) * (line - i + 1) / i

So C(line, i) can be calculated from C(line, i-1) in O(1) time

0

C(line, i)   = line! / ( (line-i)! * i! ) 

487

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9//  C++ code for Pascal's Triangle0

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9#include 48

C(line, i)   = line! / ( (line-i)! * i! ) 

441

C(line, i)   = line! / ( (line-i)! * i! ) 

67

 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1
 1 6 15 20 15 6 1

9//  C++ code for Pascal's Triangle0

//  C++ code for Pascal's Triangle0

C(line, i)   = line! / ( (line-i)! * i! ) 

1

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

29

C(line, i)   = line! / ( (line-i)! * i! ) 

401 using81

C(line, i)   = line! / ( (line-i)! * i! ) 

07

C(line, i)   = line! / ( (line-i)! * i! ) 

1

C(line, i)   = line! / ( (line-i)! * i! ) 

446

C(line, i)   = line! / ( (line-i)! * i! ) 

1

#include 99

Đầu ra

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

Thời gian phức tạp. O(n2)
Không gian phụ trợ. Ô(1)

Vì vậy, phương pháp 3 là phương pháp tốt nhất trong số tất cả, nhưng nó có thể gây tràn số nguyên cho các giá trị lớn của n vì nó nhân hai số nguyên để thu được các giá trị.  

Các biến thể của vấn đề có thể được hỏi trong các cuộc phỏng vấn

i) Tìm toàn bộ tam giác pascal như hình trên

ii) Tìm chỉ một phần tử của tam giác pascal đã cho số hàng và số cột trong thời gian O(n)

Làm cách nào để hiển thị tam giác Pascal trong C++?

Chương trình Pascal trong C là gì?

Làm cách nào để tạo tam giác Pascal trong Java?

Công thức cho tam giác Pascals là gì?