Question

How many permutations can be formed by the letters of the word ‘VOWELS’, when 

(i) there is no restriction on letters; 

(ii) each word begins with E; 

(iii) each word begins with O and ends with L; 

(iv) all vowels come together; 

(v) all consonants come together?

Answer 1

(i) There is no restriction on letters 

The word VOWELS contain 6 letters. 

The permutation of letters of the word will be 6! = 720 words. 

(ii) Each word begins with 

Here the position of letter E is fixed. 

Hence, the rest 5 letters can be arranged in 5! = 120 ways. 

(iii) Each word begins with O and ends with L 

The position of O and L are fixed.

Hence the rest 4 letters can be arranged in 4! = 24 ways.

(iv) All vowels come together 

There are 2 vowels which are O, E. 

Consider this group. 

Therefore, the permutation of 5 groups is 5! = 120 

The group of vowels can also be arranged in 2! = 2 ways.

Hence the total number of words in which vowels come together are 120 x 2 = 240 words. 

(v) All consonants come together 

There are 4 consonants V,W,L,S. consider this a group. 

Therefore, a permutation of 3 groups is 3! = 6 ways. 

The group of consonants also can be arranged in 4! = 24 ways. 

Hence, the total number of words in which consonants come together is 6 x 24 = 144 words.