Question

There are 4 badminton players, 5 volleyball players and 2 basketball players that applied to be on the athletic committee. How many different committees can be formed if the committee must consist of 3 badminton players, 2 volleyball players, and 1 basketball player.

Answer 1

For the badminton players, there are 4 available and the committee will choose 3, so the number of possible ways of choosing is a combination of 4 choose 3.

A combination of n choose p is given by:

`C(n,p)=(n!)/(p!(n-p)!)`

So, for n = 4 and p = 3, we have:

`C(4,3)=(4!)/(3!1!)=(4\cdot3!)/(3!)=4`

For the volleyball players, the committee will choose 2 from 5, so we have:

`C(5,2)=(5!)/(2!3!)=(5\cdot4\cdot3!)/(2\cdot3!)=10`

Now, for the basketball players, the committee will choose 1 from 2, so we have:

`C(2,1)=(2!)/(1!1!)=2`

Multiplying all the number of possible ways of each type of player, we have the number of different committees:

`4\cdot10\cdot2=80`

Therefore there are 80 different committees that can be formed.