Question

10 boys and 10 girls are eligible to swim in a mixed relay. In how many ways could the relay team be chosen if the team must contain 2 boys and 2 girls.
Note: the order of the swimmers on the relay matters.

Answer 1

There are 10C2 ways to choose the girls and the boys.
There are 4! ways to arrange the 4 choices. So,
10C2 * 10C2 * 4! = 90^2 * 24 = 194,400 ways

Answer 2

first choose which two girls will race (no order yet)
combinations of 10 girls 2 at a time = 10!/[ 2! (8!) ] = 45
same for the boys 45 ways
for each choice of two girls I can chose a choice of two boys
so if I choose group 1 of boys, I can choose any of the 45 groups of girls
if I choose group 2 of two boys , I can choose any of the 45 groups of girls
etc
45*45 ways
now I have four people in a room, how many ways can I order them
4 * 3 * 2 * 1 =4! = 24
so I end up with 45 * 45 * 24
I do not want to be the coach