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Question
Solve the following:
The letters of the word 'EQUATION' are arranged in a row. Find the probability that Arrangement starts with a vowel and ends with a consonant
Solution
The word EQUATION has 8 letters of which 5 (A, E, I, O, U) are vowels and 3 (Q, T, N) consonants.
The 8 letters can be arranged in 8! ways
∴ n(S) = 8!
Let B ≡ the event that arrangement starts with a vowel and ends with a consonant.
Since there are 5 vowels and 3 consonants, the first place can be arranged in 5 ways and last place can be arranged in 3 ways
After this the remaining 6 places can be arranged from remaining 6 letters in 6P6 = 6! ways.
∴ n(B) = 5 x 3 x 6!
∴ P(B) = `("n"("B"))/("n"("S"))`
= `(5 xx 3 xx6!)/(8!)`
= `(5 xx 3 xx 6!)/(8 xx 7 xx 6!)`
= `15/56`
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