Question

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

Answer 1

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 _

There are 7 places marked by “______” in which 3 vowels are to be arranged.

∴ Vowels can be arranged in ⁷P₃ 

= `(7!)/((7-3)!)`

= `(7xx6xx5xx4!)/(4!)`

= 210 ways

∴ Required number of arrangements

= 6! × 210 = 720 × 210 = 151200

∴ 151200 words can be formed if no two vowels are together.