[Math] In how many different ways can 3 men and 4 women be placed into two groups of two people and one group of three people if…

Question

I am in middle school, so simply worded answers would be super helpful!

In how many different ways can 3 men and 4 women be placed into two groups of two people and one group of three people if there must be at least one man and one woman in each group? Note that identically sized groups are indistinguishable.

As I was trying to solve this problem, I realized that the two groups of 2 people had to be made up of 1 man and 1 woman. I then multiplied c(3, 1) by c(4, 1), which equals 12. I am unsure of how to proceed from there, and I don't understand what the last note means.

Answer 1

Strategy: As you realized, there will be two groups with a man and a woman and one group with two women and one man.

1. Choose which two of the women will be in the group with three people.
2. Choose which man is in that group.
3. Choose which of the two remaining men will be in the same group with the remaining woman whose name appears first in an alphabetical list. The remaining group is then determined.

The note about identically sized groups means that the groups are not labeled. Instead, what matters is who is in which group.