A school has 8 math teachers, 6 English teachers, and 2 geography teachers. From this group, a 5 teacher committee is required. Calculate the number of ways that this committee can be formed if at least one geography teachers must be on the committee.

Question

A school has 8 math teachers, 6 English teachers, and 2 geography teachers.  From this group, a 5 teacher committee is required.  Calculate the number of ways that this committee can be formed if at least one geography teachers must be on the committee.

Reiny · Answer

The number of committees without restrictions is C(16,5) = 4368

The number of committees without any geography teachers is C(14,5) = 2002

So the number of ways with at least one geography teacher is 4368-2002 = 2366

or:
1 geography teacher : C(2,1)*C(14,4) = 2002
2 geography teachers : C(2,2)*C(14,3) = 364

sum = 2002+364 = 2366 (as above)