Question

A student council consists of 15 students.

(a)

In how many ways can a committee of five be selected from the membership of the council?

(b)

Two council members have the same major and are not permitted to serve together on a committee. How many ways can a committee of five be selected from the membership of the council?

(c)

Two council members always insist on serving on committees together. If they can't serve together, they won't serve at all. How many ways can a committee of five be selected from the council membership?

(d)

Suppose the council contains eight men and seven women.

(i)

How many committees of six contain three men and three women?

(ii)

How many committees of six contain at least one woman?

(e)

Suppose the council consists of three freshmen, four sophomores, three juniors, and five seniors. How many committees of eight contain two representatives from each class?

Answer 1

a) Total number of ways to select a committee of five out of 15 = 15C5 = 15! / (5! * (15 - 5)!) = 3003

b) Total number of ways these two members serve together = 2C2 * 13C3 = 1 * 13! / (3! * (13 - 3)!) = 286

Total number of ways to select the committee = 3003 - 286 = 2717

c) Total number of ways these two members don't serve at all = 13C5 = 13! / (5! * (13 - 5)!) = 1287

Total number of ways to select the committee = Total number of ways these two members serve together + Total number of ways these two members don't serve at all = 286 + 1287 = 1573

d) i) Number of committees with three men and three women = 8C3 * 7C3 = 56 * 35 = 1960

ii) Number of committees with at least one women = Total - number of committees with at no women

                                                                                = 15C6 - 8C6

                                                                                = 5005 - 28

                                                                                = 4977

e) Number of committees of eight contain two representatives from each class = 3C2 * 4C2 * 3C2 * 5C2

                                                                                                                             = 3 * 6 * 3 * 10

                                                                                                                             = 540