SOLUTION: Please help me solve this problem. How many ways are there to pair off eight women at the dance with eight of these 12 men? This is what I have so far, but

This is permutation problem, not a combination problem.

Here is why:

The women are W1, W2, W3, W4, W5, W6, W7, W8

Think of them lined up in that order. Not only do you
just choose 8 of the 12 men to dance with these 8 women,
but you have to place them in a certain order with these
women.

So the answer is 12P8, or 19,958,400.

[Your answer would have been correct if they weren't dancing
but were just forming a committee of 16 with 8 men and 8 women.]

Edwin