Question

The number of permutations that can be made out of the letters of the word "MATHEMATICS" When all vowels come together is:

• A. $\frac{8!4!}{2!}$
• B. $\frac{8!4!}{(2!{)}^2}$
• C. $\frac{7!4!}{2!}$
• D. $7!4!$

Answer 1

Answer: (D)

$7!4!$

In the word 'MATHEMATICS', we'll consider all the vowels AEAI together as one letter.

Thus, we have $MTHMTCS\left(AEAI\right)$.

Now, we have to arrange 8 letters, out of which M occurs twice, T occurs twice.

Number of ways of arranging these letters $=\frac{8!}{(2!)(2!)}$

Now, AEAI has 4 letters in which A occurs 2 times and the rest are different.

Number of ways of arranging these letters $=\frac{4!}{2!}$

Required number of words $=\frac{8!}{(2!)(2!)}\times \frac{4!}{2!}=\frac{8\times 7!}{(2\times 1!)(2\times 1!)}\times \frac{4!}{2\times 1!}=7!\times 4!$

Hence the correct answer is $7!4!$.