Question

Letters of the word MOTHER are arranged at random. Find the probability that in the arrangement vowels are always together.

Answer 1

There are 6 letters in the word MOTHER.

These letters can be arranged among themselves in ⁶P₆ = 6! ways.

∴ n(S) = 6!

Let event A: Vowels are always together. 

The word MOTHER consists of 2 vowels (O, E) and 4 consonants (M, T, H, R).

2 vowels can be arranged among themselves in ²P₂ = 2! ways.

Let us consider 2 vowels as one group. This one group with 4 consonants can be arranged in ⁵P₅ = 5! ways.

∴ n(A) = 2! × 5!

∴ P(A) = ${\displaystyle \frac{\text{n}\left(\text{A}\right)}{\text{n}\left(\text{S}\right)}}$

= ${\displaystyle \frac{2!\times 5!}{6!}}$

= ${\displaystyle \frac{1}{3}}$