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