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?
(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.