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

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 $\mathrm{O}$ and ends with $\mathrm{L}$;

(iv) all vowels come together;

(v) all consonants come together?

Solution:

(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 $\mathrm{O}$ and ends with $\mathrm{L}$

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 $\mathrm{O}, \mathrm{E}$.

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 $120 \times 2=240$ 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 $6 \times 24=144$ words.