In how many ways can the letter of the word $P E R M U T A T I O N$ be arranged, if the
Vowels are all together

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Solution

There are $5$ vowels and $6$ Consonants
We are to find a way such that the vowels are all together, to make this let us group all the vowels together.
Then the words can be formed in $\frac{7!}{2!} \times 5!$ ways.

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In how many ways can the letters of the word PERMUTATIONS be arranged if the

(i) words start with P and end with S, (ii) vowels are all together,

(ii) there are always 4 letters between P and S?

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