Question

Find the number of different ways of arranging letters in the word PLATOON if consonants and vowels occupy alternate positions

Answer 1

In the word 'PLATOON', the number of letters is n = 7 of which 'O' repeats twice, i.e., p = 2

∴ the total number of words formed by using the letters of the word 'PLATOON'

= `("n"!)/("p"!)`

= `(7!)/(2!)`

= `(7 xx 6 xx 5 xx 4 xx 3 xx 2!)/(2!)`

= 2520

Consonants and Vowels occupy alternate positions:

There are 4 consonants P, L, T, N, and three vowels A, O, O.

The possible arrangement in which consonants and vowels take alternate places is CVCVCVC.

Three vowels can be arranged at three places in `(3!)/(2!)` ways (∵ O repeats twice) and four consonants can be arranged at 4 places in 4! ways.

∴ the number of such arrangements

= `(3!)/(2!) xx 4!`

= `(3 xx 2!)/(2!) xx 4 xx 3 xx 2 xx 1`

= 72