I am kind of stuck in a problem with possible combinations. The problem is:
Four couples, each consisting of a man and a woman are sitting at a round table. How many possible combinations are there so that none of the couples (the man and a woman in the relationship) are sitting next to each other
I tried some things that got nowhere and now this is where I am at: Every possible combination that has nothing with them being a couple is $ 7! $. Then I wanted to count how many combinations are for the first couple to be sitting next to each other (which I think is $ 2\cdot6! $). It should be the same for every other of the three couples. But before I subtract it from the whole $ 7! $ combinations, I thought there are some possibilities I counted twice. For example, when I counted the combinations of the first couple to be sitting next to each other, I also count some of them when I count the combinations of the second couple etc. So I am kind of stuck on how to find those.
I don't know if it's a correct or the optimal way, but any comments on how to find what I am missing? Or I guess another way to approach this?