# Letters of the word MOTHER are arranged at random. Find the probability that in the arrangement starting with a vowel and end with a consonant - Mathematics and Statistics

Sum

Letters of the word MOTHER are arranged at random. Find the probability that in the arrangement starting with a vowel and end with a consonant

#### Solution

There are 6 letters in the word MOTHER, which can be arranged amongst themselves in 6P6 = 6! ways

∴ n(S) = 6! = 720

Let D ≡ the event that arrangement starts with a vowel and ends with a consonant

Vowel for the first place can be chosen in 2C1 = 2 ways.

Consonant for the last place can be chosen in 4C1 = 4 ways

The remaining 4 letters can be arranged in 4P4 = 4! = 24 ways

∴ n(D) = 2 × 4 × 24

∴ P(D) = ("n"("D"))/("n"("S"))

= (2 xx 4 xx 24)/720

= 4/15

Concept: Concept of Probability
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