<s>[INST] Choosing a 5 member team out of 12 girls and 10 boys We must choose a 5-member team from 12 girls and 10 boys. How many ways are there
to make the choice so that there are no more than 3 boys on the team?
The correct answer is $\binom{22}{5} - \binom{12}{1} \binom{10}{4} - \binom{10}{5}$.
I understand the $\binom{22}{5}$ part, but where I am confused at is the other two parts. I do not know how to get those parts. Can anyone help me understand how to get to the solution?
 [/INST] That solution proceeds by starting with $\binom{22}5$, the total number of possible $5$-person teams, and subtracting the $\binom{12}1\binom{10}4$ teams that have one girl and four boys and the $\binom{10}5$ teams that have five boys.
The problem could also be solved by noting that there are $\binom{12}2\binom{10}3$ teams with two girls and three boys, $\binom{12}3\binom{10}2$ teams with three girls and two boys, $\binom{12}4\binom{10}1$ teams with four girls and one boy, and $\binom{12}5$ teams with five girls, and calculating the sum
$$\binom{12}2\binom{10}3+\binom{12}3\binom{10}2+\binom{12}4\binom{10}1+\binom{12}5\;,$$
but it’s pretty clearly more efficient to calculate
$$\binom{22}5-\binom{12}1\binom{10}4-\binom{10}5\;.$$
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