Question

The letters of the word LOGARITHM are arranged at random. Find the probability that vowels are never together.

Answer 1

There are 9 letters in the word LOGARITHM.
These letters can be arranged among themselves in ⁹P₉ = 9! ways
∴ n(S) = 9!
There are 9 letters in the word LOGARITHM.
These letters can be arranged among themselves in 9 P9 = 9! ways
    C    C    C    C    C    C   
6 consonants create 7 gaps.
∴ 3 vowels can be arranged in 7 gaps in ⁷P₃ ways. Also 6 consonants can be arranged among themselves in ⁶P₆ = 6! ways.
∴ n(B) = 6! x `""^7"P"_3`

∴ P(B) = `("n"("B"))/("n"("S")`

= `(6! xx""^7"P"_3)/(9!)`