Question

Determine the number of arrangements of letters of the word ALGORITHM if no two vowels are together.

Answer 1

A word is to be formed using the letters of the word ALGORITHM.
There are 9 letters in the word ALGORITHM.
When no two vowels are together:
There are 6 consonants in the word ALGORITHM.
They can be arranged among themselves in ⁶P₆ i.e., 6! ways.
Let consonants be denoted by C.
     C       C       C       C       C       C      

6 consonants create 7 gaps in which 3 vowels are to arrange.
∴ 3 vowels can be filled in ⁷P₃
= `(7!)/((7-3)!)=(7xx6xx5xx4!)/(4!)` = 210 ways
∴ Total number of ways in which the word can be formed = 6! × 210 = 720 × 210 = 151200
∴ 151200 words can be formed if no two vowels are together.