How many couples end up seated together?

Question

Five couples (10 people total) are seated in a single row of 10 chairs in a completely random order. On average, how many of the 5 couples end up sitting in adjacent seats?

Accepted Answer

Whether one couple sits together shifts the odds for the others, so tracking all couples at once is messy. Put one indicator on each couple instead. Fix a couple. In a random row of n = 10 seats, treat that couple as a block: the two partners can occupy any of the 9 adjacent seat pairs, in either order. A clean way to see the probability is to place the first partner anywhere, then ask where the second lands. Whatever seat the first partner takes, the second partner is equally likely to be in any of the other 9 seats, and only the seats immediately beside the first partner count. An interior …

How many couples end up seated together?

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