Question

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

Answer 1

In the word ALGORITHM even positions are 4 and consonants are 6.

∴ number of arrangements in which 6 consonants occupy 4 even positions = ⁶P₄

Now, the remaining 5 positions can be occupied by remaining 5 letters in ⁵P₅ ways.

Hence, the total number of words in which consonants occupy even positions

= ⁶P₄ × ⁵P₅

= `(6!)/((6 - 4)!) × 5!         ...[(""^"n""P"_"r" = ("n"!)/(("n" - "r")!)),(""^"n""P"_"r" = "n"!)]`

= `(6!)/(2!) × 5!`

= `(6 × 5 × 4 × 3 × 2!)/(2!) × 5!`

=  6 × 5 × 4 × 3 × (5 × 4 × 3 × 2 × 1)

= 360 × 120

= 43200

∴ 43200 words can be formed if the consonants are at even positions.