Question

In how many ways can the letters of the word "PERMUTATIONS" be arranged if (i) words start with P and end with S, (ii) vowels are all together, (iii) there are always 4 letters between P and S?

(a) 5,040
(b) 10,080
(c) 20,160
(d) 40,320

Answer 1

The number of ways the letters of the word 'PERMUTATIONS' can be arranged if certain conditions are met is 10,080.

Step-by-step explanation:

To find the number of ways the letters of the word 'PERMUTATIONS' can be arranged given certain conditions, we can break it down into smaller steps:

1. Fix the words 'P' and 'S' at the start and end respectively. We now have 10 remaining letters to arrange.
2. Arrange the vowels (E, U, A, I, O) together. Treating them as a single unit, we now have 6 units (including consonants) to arrange.
3. Arrange the consonants (R, M, T, N) and the vowel unit together, ensuring that there are always 4 letters between 'P' and 'S'.

Using fundamental counting principle:

1. There are 2 ways to arrange the fixed letters 'P' and 'S'.
2. The vowel unit can be arranged in 5! ways, and the consonants can be arranged in 4! ways.

Combining these steps, the total number of ways is 2 * 5! * 4!. Evaluating this expression gives us the answer of 10,080. Therefore, the correct option is (b) 10,080.