A six-member research committee at a local college is to be formed having one administrator, three faculty members, and two students. There are seven administrators, 12 faculty members, and 20 students in contention for the committee. How many six-member committees are possible?

That means that, for example, a comittee of students Tre'davious and Tremaine is the same comittee as Tremaine and Tre'davious. So the combinations formula is used to solve this problem

Combinations formula:

[tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.

[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]

In this problem, we have that:

Initially, we have to find the number of groups of administrators, faculty members and students separated. Then we multiply these values to find how many six-member committees are possible.

Administrators

There are 7 administrators in contention. 1 is selected.

So there are 7 possible ways to select an administrator.

Faculty members

There are 12 faculty members in contention. 3 are selected.

[tex]C_{12,3} = \frac{12!}{3!9!} = 220[/tex]

So there are 220 ways to select groups of 3 faculty members.

Students

There are 20 students in contention. 2 are selected.

[tex]C_{20,2} = \frac{20!}{2!18!} = 190[/tex]

So there are 190 ways to select groups of 2 students.

How many six-member committees are possible?

For each administrator, there are 220 groups of faculty members.

For each group of faculty members, there are 190 groups of students.

There are 7 possible administrators

7*220*190 = 292,600

292,600 committees are possible.