A dance class consists of 22 students, 10 women and 12 men. If 5 men and 5 women are to be chosen and then paired off, how many results are possible?
Thanks!
There are $\displaystyle {{10}\choose 5}$ ways to chose the girls. And $\displaystyle {{12}\choose 5}$ ways to chose boys. In total there are $\displaystyle {{10}\choose 5}{{12}\choose 5}$ says to choose them together.
But we still are not done. That is just the number of ways of creating 10 different groups.
How many ways are there to pair them given a specific 5 boy and 5 girl group? Whatever that result is multiply it to the the number above.
For the second part we can take all 5 boys and count the number of ways of assigning each of them a girl. There are five ways to assign the first girl, 4 ways to assign the second, etc. This gives 5! ways to pair up each 5 boy 5 girl group.
Thanks! I actually had the part you helped me with. It was the second part I found tricky.
So the final answer is (12 choose 5)*(10 choose 5)*5!