Suppose there are 6 couples that go to a party, at the party they are given hats in such a way that no female can wear the same color (and males can't wear the same color).
So you have 6 pairs of hats (yellow, yellow, red, red etc.) and you want to know the amount of ways in which you can match those so that no female has the same color and no male has the same color. But the catch is NONE of the couples are allowed the same color either.
This is related to the following question: Principle of inclusion and exclusion/ Matching But there is no answer, my teacher used some sort of inclusion/exclusion method in his working out which I personally found confusing, I would appreciate either another method or an layman explanation to how you apply inclusion/exclusion here.
So a little bit of how far I've come with this, I thought that you could take the total amount of ways $ 6! \times 6! $ and then subtract/divide by the amount of ways which don't satisfy the conditions.. but I'm having trouble getting anywhere because the number changes depending on how you distribute the hats..