Question

There are 9 female board members and 21 male board members. How many ways are there to make a committee of 17 board members? How many ways are there to make a committee of 17 board members if exactly 6 must be female?

Answer 1

There are 119,759,850 ways to make a committee of 17 board members.

There are 29,628,144 ways to make a committee of 17 board members if exactly 6 must be female

Explanation:

The order that the people are chosen for the committee is not important. So we use the combinations formula to solve this question.

Combinations formula:

`C_(n,x)` is the number of different combinations of x objects from a

set of n elements, given by the following formula.

`C_(n,x) = (n!)/(x!(n-x)!)`

How many ways are there to make a committee of 17 board members?

17 from a set of 30. So

`C_(30,17) = (30!)/(17!(13)!) = 119759850`

There are 119,759,850 ways to make a committee of 17 board members.

How many ways are there to make a committee of 17 board members if exactly 6 must be female?

6 female(from a set of 9) and 11 male(from a set of 21). So

`C_(9,6)*C_(21,11) = (9!)/(6!3!)*(21!)/(11!10!) = 29628144`

There are 29,628,144 ways to make a committee of 17 board members if exactly 6 must be female