the number of ways of dividing 15 men and 15 women into 15 couples each consisting of a man and woman
Maybe it's me, but I'm thinking the above is answering a different question. Ahaok's answer would be (to me) if they had asked, how many couples can be made from 15 women and 15 men. But it's not asking that (again to me). To me they are asking, I want 15 couples; how many possible groups of 15 can I make.
Someone else want to weigh in?
I thought this was more of a permutation problem: if you think of the 15 men (or women) as a fixed position, and then you are ordering the 15 women (or men) with each ordering another possible group of 15 couples; 15 choices for the first spot, 14 choices second. . .etc. etc. Then there would be 15! possible groups of 15 couples.