Question

In how many ways can three couples be seated in a row so that each couple sits together (namely next to each other) in a circle ?

Hint : As in the general comment, recall that rotations matter. It may be useful to imagine a round table with 6 chairs, and a single mark on the table, between two specific chairs. Observe that for some configurations of the 6 people, the mark is between two different couples, denoting the mark by *, e.g., * A2A1C1C2B2B1 where A,B, and C represent the different couples, and the because of the circular arrangment B1 seats next to the *. For other configurations, the mark separates two members of the same couple for example * A1C1C2B2B1A2, where A2 seats next to the *. You can count the number of configurations of each type.

Answer 1

We know, in case of circular permutation around a table with n individuals, number of possible permutations = (n-1)!

Here, we can treat a couple as one individual as they seat together.

So, there are 3 couples any they can be assigned together in (3-1)! = 2! ways.

Further for every couple, there are 2 individuals and they can be arranged in 2! ways in there allotted 2 seats.

Thus total number of possible ways = 2!*(2!)3 = 16