Question

A school dance committee is to consist of 2 freshmen, 3 sophomores, 4 juniors, and 5 seniors. If5 freshmen, 9 sophomores, 7 juniors, and 7 seniors are eligible to be on the committee, in how many ways can the committee be chosen? ​

Answer 1

17,640 ways

Explanation:

This is a problem in combinatorics which tells you in how many ways you can pick r items from a total of n items.

The notation for this is
`{n \choose r}` also written as C(n, r)

`\rm C(n, r)\;= { \binom{n}{r}} }\; is \; given\; by \; the \; formula\; \; \frac {n!}{r!(n-r)!}\\where n! = factorial of n = n * (n-1) * (n-2) * ........* 3 * 2 * 1`

Let's deal with each of the choices

2 freshmen out of 5 freshmen.
This is C(5, 2)

`= = (5!)/(( 2! (5 - 2)! )) = = (5!)/(2! * 3! ) = 10`

3 sophomores out of 9 sophomores

= C(9, 3)

`= (9!)/(( 3! (9 - 3)! )) = = (9!)/(3! * 6! ) = 84`

4 juniors out of 7 juniors

C(7, 4)

`= (7!)/(( 7! (7 - 7)! )) = (7!)/(7! * 0! ) =1`

5 seniors out of 7 seniors
C(7, 5)

`= (7!)/(( 5! (7 - 5)! )) = (7!)/(5! * 2! ) = 21`

So total number of ways in which we can fill this committee
= 10 x 84 x 1 x 21

= 17,640 ways