Question

The number of ways in which all the letters of the word "INTEGRATION" can be arranged so that all vowels are always in the beginning of the word is

• A. $\frac{6!5!}{(2!{)}^3}$
• B. $\frac{7!4!}{2!}$
• C. $7!\cdot 4!$
• D. $\frac{7!}{(2!{)}^2}$

Answer 1

The correct option is A $\frac{6!5!}{(2!{)}^3}$

A possible arrangement for the given condition is

IEAIONTGRTN

Now the $5$ vowels can be permuted in the first five positions and the consonants can be permuted in the last $6$ positions

Also, two I's two N's and two T's are repeated.

Therefore, total number of permutations $=\frac{6!5!}{2!2!2!}$

Correct option is A.