Question

In how many ways can you arrange the members of two committees of 11 males and 11 females around a circular table if the only thing that matters is who the neighbors of each committee member are on each side (having person 1 on the right and person 2 on the left is different from having person 1 on the left and person 2 on the right), and not who sits in which chair if:

1. There are no restrictions

2. The two leaders of the committee members cannot sit next to each other

3. Two males and two females cannot be seated next to each other (every other gender)

Answer 1

1. Let us first label the chairs clockwise. In this case we have 22! ways to place the members on the labelled chairs. But the labelling can be done in 22 ways(1st seat can be chosen in 22 ways and the rest of the labels are assigned automatically after that). Hence the total number of ways to place the members on unlabelled chairs(that is it doesn't matter who sits on which chair) is given by
2. Place one leader on any chair(it doesn't matter which as only the relative order has to be considered). The second leader has 19 options for the chair to sit on, the rest of the members can be arranged in 20! ways. So the total number ways to place members in this case is .
3. In this case the male and female members alternate. Therefore we first place the men on the chairs, leaving gaps between them. This is equivalent to the first part with only 11 members and 11 chairs. Hence total number of ways to place the men is 10!. Now, while placing the women, note that since the men are already sitting on the chairs, this time it matters which woman sits on which chair (not just the relative ordering with respect to other women). So the first woman has 11 options, second woman has 10 options and so on. Hence after placing the men there are 11! ways to place the women. Hence the total number of ways place the members in this case is .