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

Explanation

To solve the problem of arranging the letters of the word "PERMUTATIONS" under the given conditions, we will break it down step by step.

### Step 1: Understanding the word "PERMUTATIONS"
The word "PERMUTATIONS" consists of 12 letters in total, with the following breakdown:
- Letters: P, E, R, M, U, T, A, T, I, O, N, S
- Total letters: 12
- Repeated letters: T (2 times)

...