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

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#### Solution

There are 6 letters in the word MOTHER, which can be arranged amongst themselves in ^{6}P_{6} = 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 ^{2}C_{1} = 2 ways.

Consonant for the last place can be chosen in ^{4}C_{1} = 4 ways

The remaining 4 letters can be arranged in ^{4}P_{4} = 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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