How many permutations can be made with letters of the word MATHEMATICS in how many of them vowels are together?

Distinguishable Ways to Arrange the Word MATHEMATICS
The below step by step work generated by the word permutations calculator shows how to find how many different ways can the letters of the word MATHEMATICS be arranged.

Objective:
Find how many distinguishable ways are there to order the letters in the word MATHEMATICS.Step by step workout:
step 1 Address the formula, input parameters and values to find how many ways are there to order the letters MATHEMATICS.
Formula:
nPr =n!/(n1! n2! . . . nr!)Input parameters and values:
Total number of letters in MATHEMATICS:
n = 11

Distinct subsets:
Subsets : M = 2; A = 2; T = 2; H = 1; E = 1; I = 1; C = 1; S = 1;
Subsets' count:
n1(M) = 2, n2(A) = 2, n3(T) = 2, n4(H) = 1, n5(E) = 1, n6(I) = 1, n7(C) = 1, n8(S) = 1

step 2 Apply the values extracted from the word MATHEMATICS in the (nPr) permutations equation
nPr = 11!/(2! 2! 2! 1! 1! 1! 1! 1! )

= 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11/{(1 x 2) (1 x 2) (1 x 2) (1) (1) (1) (1) (1)}

= 39916800/8

= 4989600
nPr of word MATHEMATICS = 4989600

Hence,
The letters of the word MATHEMATICS can be arranged in 4989600 distinct ways.

Apart from the word MATHEMATICS, you may try different words with various lengths with or without repetition of letters to observe how it affects the nPr word permutation calculation to find how many ways the letters in the given word can be arranged.

(b) 120960

There are 11 letters in the word “MATHEMATICS” out of which 4 are vowels and the rest 7 are consonants. 

Let the four vowels be written together. A A E I  M, T, H, M, T, C, S 

Consider the four vowels as one as unit, then these 8 letters (7 consonants and the vowel unit) can be permuted in \(\frac{8!}{2!2!}\) = 10080 ways. (There are two pairs of same letters AA and MM) 

Corresponding to each of these permutations, the 4 vowels can be arranged among themselves in \(\frac{4!}{2!}\) = 12 ways. 

∴ Required number of word in which vowels occur together = \(\frac{8!}{2!2!}\times\frac{4!}{2!} = 10080\times12\) = 120960.

Mathematic can be arranged in 453,600 different ways if it is ten letters and only use each letter once. Assuming all vowels will be together 15,120 arrangements.

Requires work with permutations and factorials. Factorial is written as '!' . Factorial is the multiplication of all it lower terms.
Eg 10! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1

Mathematic has ten letters and as such can be arranged 10! ways. As it has repeating letters you divide by this repetitions.

As such it becomes #(10!)/(2!*2!*2!)# . This equation equals 453 600

For the second part it should be treated as two parts. All the vowels are grouped together so there are effectively only 7 letters left.

This you would write as #(7!)/(2!*2!)# (Lose a 2! as the repeating vowel is not counted.) Then for the vowel, it is #(4!)/(2!)#.
You now multiply these together to get your answer of 15120

In how many words can the letters of word ‘Mathematics’ be arranged so that (i) vowels are together (ii) vowels are not together

Answer

Verified

Hint: First find the number of ways in which word ‘Mathematics’ can be written, and then we use permutation formula with repetition which is given as under,
Number of permutation of $n$objects with$n$, identical objects of type$1,{n_2}$identical objects of type \[2{\text{ }} \ldots \ldots .,{\text{ }}{n_k}\]identical objects of type $k$ is \[\dfrac{{n!}}{{{n_1}!\,{n_2}!.......{n_k}!}}\]

Complete step by step solution:
Word Mathematics has $11$ letters
\[\mathop {\text{M}}\limits^{\text{1}} \mathop {\text{A}}\limits^{\text{2}} \mathop {\text{T}}\limits^{\text{3}} \mathop {\text{H}}\limits^{\text{4}} \mathop {\text{E}}\limits^{\text{5}} \mathop {\text{M}}\limits^{\text{6}} \mathop {\text{A}}\limits^{\text{7}} \mathop {\text{T}}\limits^{\text{8}} \mathop {\text{I}}\limits^{\text{9}} \mathop {{\text{ C}}}\limits^{{\text{10}}} \mathop {{\text{ S}}}\limits^{{\text{11}}} \]
In which M, A, T are repeated twice.
By using the formula \[\dfrac{{n!}}{{{n_1}!\,{n_2}!.......{n_k}!}}\], first, we have to find the number of ways in which the word ‘Mathematics’ can be written is
$
  P = \dfrac{{11!}}{{2!2!2!}} \\
   = \dfrac{{11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}}{{2 \times 1 \times 2 \times 1 \times 2 \times 1}} \\
   = 11 \times 10 \times 9 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \\
   = 4989600 \\
 $
In \[4989600\]distinct ways, the letter of the word ‘Mathematics’ can be written.

(i) When vowels are taken together:
In the word ‘Mathematics’, we treat the vowels A, E, A, I as one letter. Thus, we have MTHMTCS (AEAI).
Now, we have to arrange letters, out of which M occurs twice, T occurs twice, and the rest are different.
$\therefore $Number of ways of arranging the word ‘Mathematics’ when consonants are occurring together
$
  {P_1} = \dfrac{{8!}}{{2!2!}} \\
   = \dfrac{{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}}{{2 \times 1 \times 2 \times 1}} \\
   = 10080 \\
 $
Now, vowels A, E, I, A, has $4$ letters in which A occurs $2$ times and rest are different.
$\therefore $Number of arranging the letter
\[
  {P_2} = \dfrac{{4!}}{{2!}} \\
   = \dfrac{{4 \times 3 \times 2 \times 1}}{{2 \times 1}} \\
   = 12 \\
 \]
$\therefore $Per a number of words $ = (10080) \times (12)$
In which vowel come together $ = 120960$ways

(ii) When vowels are not taken together:
When vowels are not taken together then the number of ways of arranging the letters of the word ‘Mathematics’ are
$
   = 4989600 - 120960 \\
   = 4868640 \\
 $

Note: In this type of question, we use the permutation formula for a word in which the letters are repeated. Otherwise, simply solve the question by counting the number of letters of the word it has and in case of the counting of vowels, we will consider the vowels as a single unit.

How many permutations can be made from the letters of the word MATHEMATICS if the vowels are to be together?

How many permutations can be made with the letters of the word MATHEMATICS?

How many words can be made from the word MATHEMATICS in which vowels are together?

How many ways can MATHEMATICS be arranged so that the vowels come together?