I have come across the following counting poblem and not sure if I've followed the right path to finding the answer. The problem is: There are 5 couples. The host sits at the head of the table and the hostess at the opposite head of the table. Its a rectangluar table with with 4 seats either side of the table between the 2 heads of the table. No man or woman can sit next to eachother. Each member of a couple must be on the other side to their partner.
My method was to seat the women first in 4! ways. Then on the one side of the table there is only a choice of 2 men since there cant be the men who are the partners of the women on that side-that can be done in 2! ways. Then is did exactly the same for the other side. This yields: 4!x 2!x 2!
I was unsure whether I was missing a 2!-whether u can rearrange the heads of the table in 2! ways. Or whether Im following the wrong route?