Couples around a round table: how many sit together?

Question

Six couples (12 people total) are seated at random around a circular table with 12 seats. On average, how many of the 6 couples end up sitting in adjacent seats?

Accepted Answer

Put one indicator on each couple and add, so the dependence between couples never has to be handled. Fix a couple. Around a circular table of n = 12 seats, every seat has exactly 2 neighbors. Seat one partner anywhere; the other partner is equally likely to be in any of the remaining n - 1 = 11 seats, and exactly 2 of those are next to the first partner. So P(	ext{a given couple is adjacent}) = \frac{2}{n - 1} = \frac{2}{11}. Let Xi = 1 if couple i is adjacent; E[Xi] = 	frac{2}{11}. With 6 couples, E[X] = 6 	imes \frac{2}{11} = \frac{12}{11} \approx 1.091. The circle gives a slightly large…

Couples around a round table: how many sit together?

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