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One Indian and four American men and their wives are to be seated randomly around a circular table. Then, the conditional probability that the Indian man is seated adjacent to his wife given that each American man is seated adjacent to his wife is

(a) \(\frac{1}{2}\) 

(b) \(\frac{1}{3}\) 

(c) \(\frac{2}{5}\) 

(d) \(\frac{1}{5}\)

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Answer: (C) =\(\frac{2}{5}\)

Let A: Event that Indian man is seated adjacent to his wife. 

Let B: Event that each American is seated adjacent to his wife. 

Consider each couple as one entity. Thus, there are 5 entities to be arranged and husbands and wife can interchange their seats in 2! ways.

∴ \(P(A\,\cap\,B)\) = \(\frac{4!(2!)^5}{9!}\)

Next, consider each American couple as an entity. Thus, there are 6 entities to be arranged including the Indian and his wife.

∴ P(B) = \(\frac{5!(2!)^4}{9!}\)

∴ P(A/B) = \(\frac{P(A\,\cap\,B)}{P(B)}\) = \(\frac{4!(2!)^5}{5!(2!)^4}\) = \(\frac{2}{5}\)

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