A computer programming team has 13 members.
a. How many ways can a group of seven be chosen to work on a project?
b. Suppose seven team members are women and six are men.
(i) How many groups of seven can be chosen that contain four women and three men?
(ii) How many groups of seven can be chosen that contain at least one man?
(iii) How many groups of seven an be chosen that contain at most three women?
c. Suppose two team members refuse to work together on projects. How many groups of seven can be chosen to work on a project?
d. Suppose two team members insist on either working together or not at all on projects. How many groups of seven can be chosen to work on a project?

(a)

To calculate:

The number of groups of 7 that can be chosen from 13 members.

Given information:

There are 13 members in the team and only 7 members should be chosen.

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 the combinations of groups, the order is not considered. Also, no repetitions in the groups. Hence, we can substitute n=13 and r=7 into the formula

(b)

To calculate:

The number of groups of 7 that satisfies following conditions.

1. Four women and three are men in the group.
2. At least one man in the group.
3. At most three women in the group.

(c)

To calculate:

The number of different groups of seven members without two specific team members together.

(d)

To find:

The number of groups of 7 members with either both two members are together or both are not together.