Answer :
In order to find the number of different ways the committee can be made, first let's calculate the possibilities for the teachers.
Since we have 8 teachers and need to choose 3, we have a combination of 8 choose 3.
A combination of n choose p is calculated with the formula below:
[tex]C(n,p)=\frac{n!}{p!(n-p)!}[/tex]So, for n = 8 and p = 3, we have:
[tex]C(8,3)=\frac{8!}{3!(8-3)!}=\frac{8!}{3!\cdot5!}=\frac{8\cdot7\cdot6\cdot5!}{3\cdot2\cdot5!}=56[/tex]Now, for the students, we have 15 students and need to choose 6, so we have a combination of 15 choose 6:
[tex]C(15,6)=\frac{15!}{6!9!}=\frac{15\cdot14\cdot13\cdot12\cdot11\cdot10\cdot9!}{6\cdot5\cdot4\cdot3\cdot2\cdot9!}=5005[/tex]Multiplying both numbers of possibilities, we have the final result:
[tex]56\cdot5005=280280[/tex]Therefore there are 280,280 different ways of making the committee.