Question

6 teachers and 12 students volunteer for a committee to discuss extra curricular activities. How many committees of 5 people can be made if there muwt be at least one teacher and at least one student on the committee

Answer 1

To find the total number of committees, we need to consider the different cases.

Case 1: 1 teacher and 4 students
We can choose 1 teacher out of 6 teachers in $\binom{6}{1}$ ways and 4 students out of 12 students in $\binom{12}{4}$ ways.
So the total number of committees with 1 teacher and 4 students is $\binom{6}{1} \cdot \binom{12}{4}$.

Case 2: 2 teachers and 3 students
We can choose 2 teachers out of 6 teachers in $\binom{6}{2}$ ways and 3 students out of 12 students in $\binom{12}{3}$ ways.
So the total number of committees with 2 teachers and 3 students is $\binom{6}{2} \cdot \binom{12}{3}$.

Case 3: 3 teachers and 2 students
We can choose 3 teachers out of 6 teachers in $\binom{6}{3}$ ways and 2 students out of 12 students in $\binom{12}{2}$ ways.
So the total number of committees with 3 teachers and 2 students is $\binom{6}{3} \cdot \binom{12}{2}$.

Case 4: 4 teachers and 1 student
We can choose 4 teachers out of 6 teachers in $\binom{6}{4}$ ways and 1 student out of 12 students in $\binom{12}{1}$ ways.
So the total number of committees with 4 teachers and 1 student is $\binom{6}{4} \cdot \binom{12}{1}$.

Adding all the cases together, the total number of committees is
$\binom{6}{1} \cdot \binom{12}{4} + \binom{6}{2} \cdot \binom{12}{3} + \binom{6}{3} \cdot \binom{12}{2} + \binom{6}{4} \cdot \binom{12}{1}$
$= 6 \cdot 495 + 15 \cdot 220 + 20 \cdot 66 + 15 \cdot 12$
$= 2970 + 3300 + 1320 + 180$
$= \boxed{7830}$ committees.