In how many ways can a committee be selected? Given: 2 teachers, 7 boys, 4 girls
The question is:
"If the committee (5 persons) must include at least one teacher and two boy students, in how may ways can the committee be selected?"
My question is if I can include more than two boys, or must there be exactly 2 boys.
Thank you.
 A: Ask the author of the problem, whether "at least two boys" or "exactly two boys" is meant. MSE cannot help you in this dilemma.
In the following I'm treating the "at least two boys" interpretation.
We can take both teachers, $b\in\{2,3\}$ boys and $g=3-b$ girls in
$${7\choose 2}{4\choose 1}+{7\choose 3}{4\choose0}=119$$
ways, and we can take $1$ teacher, $b\in\{2,3,4\}$ boys and $g=4-b$ girls in
$$2{7\choose 2}{4\choose2}+2{7\choose3}{4\choose1}+2{7\choose4}{4\choose0}=602$$
ways, giving a total of $721$.
A: Assuming that at least two boys are required: figure out how many different ways $1$ required teacher slot can be filled from $2$ different teachers, multiply that by the number of ways the $2$ slots for boys can be filled from $7$ different boys, then multiply that by the number of ways the remaining two slots can be filled from $(2 - 1)$ teachers plus $(7 - 2)$ boys plus $4$ girls. If exactly two boys are required, then disregard the boys when computing who can fill the last two spots.
