Question

7.A group has 9 women and 7 men.

d. Suppose 2 members of the group refuse to work together. How many subgroups of 5 can be chosen

8. In how many ways can 16 people be seated:

a. In a row, if 4 of the 16 do not want to sit next to one another

b. In a row, if 3 of the 16 must be seated next to one another

c. In a circle, if 3 of the 16 must be seated next to one another.

d. In a circle, if 4 of the 16 do not want to sit next to one another.

Answer 1

I'll tackle 7d.

Out of 7 people, two do not want to work together.

Method 1:
Calculate all possible ways (5 out of 7), and subtract those that include the two particular persons (3 out of 5).
C(7,5)-C(5,3)
=7!/(5!2!)+5!/(3!2!)
=11

Method 2:
Make 3 cases,
1. without A, nor B : 1 way
2. with only A and not B:
C(5,4)=5
3. with only B and not A:
C(5,4)=5
Total: 1+5+5=11