# 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.

• Exactly two boys – Upstart Mar 1 '16 at 16:26
• Resolve both situations. I tend to see here "at least two boys". – Masacroso Mar 1 '16 at 16:39
• inclusion and exclusion works here – ThunderWiring Mar 1 '16 at 16:57
• The question is not clearly worded, but I tend to agree with Masacroso. – true blue anil Mar 1 '16 at 17:24
• I agree with Masacroso that it's probably "at least two boys", but ask for clarification if possible. If not, work the problem for both cases, they're not that much different. – DylanSp Mar 1 '16 at 17:45

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 $721$.
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.