EXAMPLE 3.3.8

Six Democrats and 5 Republicans are in a jail cell. Either 3 Democrats or 3 Republicans will be chosen to pick up trash alongside the highway. How many different 3-person groups are possible?

A. 30

B. 200

C. 7200

D. 180

SOLUTION

Since the group will consist of either 3 Democrats or 3 Republicans, the total number of groups is the number of 3-Democrat groups plus the number of 3-Republican groups.

Number of 3-Democrat groups

We are choosing three Democrats from the set of six Democrats; since they are all receiving the same treatment, the order in which they are chosen or listed is not important. Thus, the number of 3-Democrat groups is C(6,3).

C(6, 3) = 6 factorial divided by (3 factorial times 3 factorial) = 20.

Number of 3-Republican groups

We are choosing three Republicans from the set of five Republicans; since they are all receiving the same treatment, the order in which they are chosen or listed is not important. Thus, the number of 3-Republican groups is C(5,3).

C(5, 3) = 5 factorial divided by (2 factorial times 3 factorial) = 10.

The total number of three person groups is

20 + 10 = 30