A student council consists of 15 students.

a. In how many ways can a committee of six 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 six 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 six 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?

(a)

To calculate:

The number of committees with 6 members that can be formed from 15 students.

Given information:

There are 15 students in the student council and the committee consists 6 members.

Formula used:

The number of r− combinations that can be chosen from a set of n elements is given by (nr).

(nr)=n!(n−r)!×r!

Calculation:

When selecting he committee, the order of the members is not considered. Also, no repeated individuals in the combinations. Hence, we can substitute n=15 and r=6 into the formula

(b)

To calculate:

The number of committees of 6 members without two specific members without together.

(c)

To calculate:

The number of committees can be chosen with two specific council members who serve together.

(d)

To find:

1. The number of committees with three men and three women.
2. The number of committees with at least one women in the 6 members.

(e)

To find:

The number of committees of eight members that contain two students from each year.