Combinatorics Problem: Choosing Couples in a Dance Class with 22 Students

[arsato]

Homework Statement

A dance class consists of 22 students, of which 10 are women and 12 are men. If 5 men and 5 women are to be chosen and then paired off, how many results are possible?

Homework Equations

The Attempt at a Solution

We can say that for the first couple, there are a pool of 12 possible men to choose from and a pool of 10 possible women to choose from. So there are 12x10=120 possible couples. For the second couple, there are a pool of 11 men and a pool of 9 women. So 9x11=99. and so on until we get to the fifth couple (8x6=48). Then we add all the numbers together (120+99+80+63+48=410 possible 5 couple combinations). Is this the correct reasoning? I feel I'm missing something here.

 

[Moos3d]

The reasoning is not correct. Consider a much simpler problem: you have two men, two women and you want to pair them into couples, how many ways can you do this? If we let the men be A and B, and the women be 1 and 2, then the only two distinct pairs of couples we have are:

(A,1) and (B,2)
(A,2) and (B,1)

so the answer for this is two. But by your previous reasoning we would conclude that the answer is 2*2 + 1*1 = 5.

 

[arsato]

Okay, so what would be a correct approach?

 

[Outlined]

Given 10 men, how many ways can you pick 5 men out of it?
Given 12 women, how many ways can you pick 5 women out of it?
Given 5 men and 5 women, how many couples can you make?