How many ways to assemble a team of 5 out of 15 girls and 10 boys, if the team must contain at least two boys and two girls?
Why is it wrong to count the following way(?):
- Choose 2 boys $ = {10 \choose 2}$
- Choose 2 girls $ = {15 \choose 2}$
- Choose one more boy/girl from the rest $ = 21$
$ {10 \choose 2}{15 \choose 2}21 = 99225$
I quiet sure that this count has duplicates, if so, how do I eliminate them?
The correct answer is: ${10 \choose 2}{15 \choose 3}+{10 \choose 3}{15 \choose 2}=33075 $