Homework Help: Number of groups of dance couples from pool of M,F

1. Oct 3, 2016

hotvette

1. The problem statement, all variables and given/known data
How many groups of 5 dances couples can be formed from a pool of {12M, 10F}?

2. Relevant equations
$${}^n\!P_k = \frac{n!}{(n-k)!} \\ {}^n\!C_k = \frac{n!}{k!(n-k)!}$$

3. The attempt at a solution
We were shown one solution in class which is to find the number of groups of 5M that can be formed from {12M} multiplied by the number of groups of 5F that can be formed from {10F} times the number of groups of 5 couples can be formed from {5M, 5F}:

$${}^{12}\!C_5 \cdot {}^{10}\!C_5 \cdot 5! = 23,950,080$$

I thought of alternative approach: find the number of unique couples that can be formed from {12M, 10F} and from that pool find out how many groups of 5 can be formed:

$$n = 12 \cdot 10 = 120 \\ {}^{120} C_5 = 190,578,024$$

What is wrong with the 2nd approach?

2. Oct 3, 2016

andrewkirk

The 120 candidate couples includes (June + Dave) and (June + Remy). But at most one of those couples can be in our set of five, as June can only dance with one person at a time. The second method allows two of the five couples to be those two.

3. Oct 3, 2016

hotvette

Ah, thanks. Is there any way to correct for the double counting, or is this approach a non starter?