Question

In how many ways can the letters of the word 'INTERMEDIATE' be arranged so that :

(i) the vowels always occupy even places ?

(ii) the relative order of vowels and consonants do not alter ?

Answer 1

INTERMEDIATE
I = 2 times, T= 2 times, E=3 times, N, R, M, D, A Number of letters = 12

(i) There are 6 vowels. They occupy even places 2nd, 4th, 6th„ 8th, 10th, 12th. After there six there are six places and 5 letters, T is 2 times.

So, number of ways for consonants = $\frac{6!}{2!}$

The total number of ways when vowels occupy even places = $\frac{6!}{2!}\times \frac{6!}{2!3!}$

= $\frac{6\times 5\times 4\times 3\times 2\times 6\times 5\times 4\times 3\times 2}{2\times 2\times 3\times 2}$

= 21600

(ii) Number of ways such that relative order of vowels and consonents do not alter

= $\frac{6!}{2!\times 3!}\times \frac{6!}{2!}$

= 21600

Required number of ways = 21600