Question

Find the number of different ways of arranging letters in the word PLATOON if (a) the two O’s are never together. (b) consonants and vowels occupy alternate positions

Answer 1

There are 7 letters in the words PLATOON in which ‘O’ repeat 2 times.

(a) When the two O’s are never together.

Let us arrange the other 5 letters first, which can be done in 5! = 120 ways.

The letters P, L, A, T, N create 6 gaps, in which O’s are arranged.

Two O’s can take their places in ⁶P₂  ways.

But ‘O’ repeats 2 times.

∴ Two O’s can be arranged in \(\frac{^6P_2}{2!}\)

= 15 ways

∴ Required number of arrangements = 120 × 15 = 1800

(b) When consonants and vowels occupy alternate positions. There are 4 consonants and 3 vowels in the word PLATOON.

∴ At odd places, consonants occur and at even places, vowels occur. 4 consonants can be arranged among themselves in 4! ways.

3 vowels in which O occurs twice and A occurs once.

∴ They can be arranged in ways. Now, vowels and consonants should occupy alternate positions.

∴ Required number of arrangements = 4! × = 4 × 3 × 2 × \(\frac{3\times2!}{2!}\) = 72