MGF 1107/Class Examples/ Combinatorics

1. A senator must hire a new chief-of-staff and a new press secretary. If there are 5 applicants for chief-of-staff and 3 applicants for press secretary, how many different ways can these two positions be filled?

2. All of a sudden, the senator’s director of legislative affairs quits. If there are two applicants for this position, in how many ways can the senator fill the three positions?

3. A voter has to rank 5 alternatives. How many different preference lists are possible?

4. Simplify:

a) 7! b) c)

5. How many permutations are there of the letters ABCDE?

6. A congressman plans to nominate 3 high school seniors to the U.S. Military Academy. West Point asks the congressman to rank the three nominees in order of preference. If the congressman is considering 7 candidates, how many ways can he choose 3 and rank them in order of preference?

7. Find:

a) b) c)

8. How many subsets does {a, b, c, d, e} have?

9. a) How many subsets does {a, b, c, d} have?

b) How many subsets does {a, b, c, d} have that contain exactly 3 elements?

10. How many 4-element subsets does a 9-element set have?

11. From a committee of 9 members, 3 must be chosen for a subcommittee. In how many different ways can this be done?

12. From a committee of 9 members, 3 must be chosen for the positions of chair, vice-chair, and recording secretary. In how many different ways can this be done?

13. An election using the method of sequential pairwise voting has 4 alternatives: A, B, C, and D. How many different agendas are possible?

14. To see if an election with 7 alternatives has a Condorcet winner, we could look at every possible head-to-head match up. How many such match ups are there?

15. An election using the method of sequential pairwise voting has 7 alternatives: A, B, C, D, E, F and G. How many different agendas are possible that have B as the third item on the agenda?

16. A governor needs to appoint 6 members to the state’s Board of Education. If he is considering 7 whites and 5 blacks for the openings, in how many different ways can he choose 4 whites and 2 blacks for the Board?