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

detailed solution

Correct option is C

Let E1 be the event that the Indian man is seated  adjacent to his wife and E2, be the even that each Americanman is seated adjacent to his wife. Then, PE1/E2=PE1∩E2PE2Now, E1∩E2 is the event that all men are seated adjacent to their wives.Therefore, we can consider the 5 couples as single-single objects which can be arranged in a circle in 4! ways. But foreach couple, husband and wife can interchange their places in 2! ways.Hence, the umber of ways when all men are seated adjacent to their wives is 4!×(2!)5. Also in all, 10 persons can be seatedin a circle in 9 ! ways. Therefore,PE1∩E2=4!×(2!)59!Similarly, if each American man is seated adjacent to his wife, considering each American couple as single object and Indian woman and man as separate objects, there are 6 different objects which can be arranged in a circle in 5 ! ways. Also for each American couple, husband and wife can interchange their places in 2! ways.So, the number of ways in which each American man is seated adjacent to his wife is 5!×(2!)4. Therefore,PE2=5!×(2!)49!Hence,PE1/E2=4!×(2!)5/9!5!×(2!)4/9!=25