Combinatorial enumeration in society

In casual conversation, it is normal to automatically pick up small clues to figure out people's relationship to one another. A glance of mutual understanding, a disparaging remark often suffice to identify couples. How much information is needed to figure out who is paired with whom?

In a group of six people, three men and three women, it usually doesn't take long to figure it out. Indeed, there are only 3.2.1=6 possibilities. But recently, conversing with a group of gay men, I was surprised to find that I couldn't do inferences as usual. Why not? It's simply a matter of combinatorial enumeration: with six men there are 5.3.1=15 possibilities instead of 6.

In general, assuming perfect matchings, with k men and k women there are k(k-1)(k-2)... 1 possibilities. With 2k gay men, there are (2k-1)(2k-3)...3.1 possibilities.