3 married couples go out for dinner and are seated randomly around a circular table for six people. What is the probability that exactly and only 1 of the 3 couples are seated together?

You can solve 3 Couples in a Circle -- II here

The answer is 0.4.

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Fix a couple seated together. They can change places (2 x). The other couples can only sit one seat apart. They can exchange places (2 x2x). they can exchange seats with the other couple (2x). Thus there are 2x2x2x2= 16 positions when one couple is fixed. Since fixing other two couples will lead to different seating, the total number of ways is 16x3 = 48.

6 people can sit in a circle in 5! ways = 120.

Probability of exactly one couple seated together = 48/120 = 0.4