Question

Determine the number of arrangements of letters of the word ALGORITHM if consonants are at even positions.

Answer 1

A word is to be formed using the letters of the word ALGORITHM.
There are 9 letters in the word ALGORITHM.
When consonants are at even positions:
There are 4 even places and 6 consonants in the word ALGORITHM.
1^st, 2^nd, 3^rd, 4^th even places are filled in 6, 5, 4, 3 way respectively.
∴ The number of ways to fill four even places by consonants = 6 × 5 × 4 × 3 = 360
Remaining 5 letters (3 vowels and 2 consonants) can be arranged among themselves in ⁵P₅ i.e., 5! ways.
∴ Total number of ways the words can be formed in which even place is occupied by consonants
= 360 × 5! = 360 × 120 = 43200
∴ 43200 words can be formed if even positions are occupied by consonants.