Question

A committee consisting 4 of faculty members and 5 students is to be formed. Every committee position has the same duties and voting rights. There are 9 faculty members and 14 students eligible to serve on the committee. In how many ways can the committee be formed?

Answer 1

The committee can be formed by 252252 ways

SOLUTION:

Given, a committee consisting 4 of faculty members and 5 students is to be formed.

There are 9 faculty members and 14 students eligible to serve on the committee.

We need to find In how many ways can the committee be formed.

Every committee position has the same duties and voting rights. So, we need not consider any order and we just need to find the total possible combinations.

Total possible combinations = combinations possible for teachers
`*` combinations possible for students.

Now, combinations possible for teachers = 4 out of 9 teachers

`\begin{array}{l}{=^(9) \mathrm{C}_(4)} \\\\ {=(9 !)/((9-4) ! 4 !)} \\\\ {=(9 !)/(5 ! * 4 !)} \\\\ {=(9 * 8 * 7 * 6 * 5 !)/(5 ! * 3 * 2 * 1)} \\\\ {=(3024)/(24)}\end{array}`

Combinations possible for teachers = 126

Now, combinations possible for students = 5 out of 14 teachers

`\begin{array}{l}{=^(14) \mathrm{C}_(5)} \\\\ {=(14 !)/((14-5) ! 5 !)} \\\\ {=(14 !)/(9 ! * 5 !)} \\\\ {=(14 * 13 * 12 * 11 * 10)/(5 * 4 * 3 * 2 * 1)} \\\\ {=(240240)/(120)}\end{array}`

Combinations possible for students = 2002

Now, Total possible combinations = 126
`*` 2002

Total possible combinations = 252252.

Hence, totally 252252 possible ways are there to form the committee.