Question

In how many ways can four men and four women be seated around a circular table if each man must be flanked by two women?

Answer 1

144

Explanation:

If we consider the question for a second, with each man being flanked by 2 women, the seating arrangement must be set up as MWMWMWMW (around a table).

Let's put a M in seat 1, then W must be in seats 8 and 2, and so on. This arrangement gives `4!xx4!`.

We also need to account for the fact that we can start at seat 1 being a W, and putting M in seats 8 and 2, and so on. And so we get `2xx4!xx4!`

The last thing we need to account for is that we're seating around a table. Unlike a row of seats, there is no distinguishable "first seat". And so to account for that, we divide by the number of seats, giving:

`(2xx4!xx4!)/8=(2xx24xx24)/8=144`

Does this make sense?

To check, let's put the M in seats 1, 3, 5, 7 and arrange the women around them - this can be done in `4! =24` ways. We can then arrange the men into a different order - but to avoid replication, we fix one man, let's call him `M_1`, in a seat and then arrange the other men around him. That can be done in `3! = 6` ways. And for each 6 arrangements of the M, we have 24 arrangements of W, giving:

`24xx6=144`