MGF 1107/ Combinatorics/ Homework

1. Simplify:

a) 5! b) 0! c) d) e) Chapter 11, exercise #17

2. Six candidates are in an election. How many different preference lists are possible?

3. How many subsets does {A, B, C, D, E, F} have?

4. Of the subsets in number 3, how many have 5 elements?

5. A committee on foreign affairs consists of 11 members. A subcommittee of 5 members needs to be selected to deal with Latin American affairs. In how many different ways can this subcommittee be selected?

6. Senator Graham plans a campaign swing to include stops in Miami, Tampa, Orlando, and Jacksonville. One possible itinerary has him first going to Tampa, then Miami, then Jacksonville, and finally Orlando. How many possible itineraries are there in all?

7. Governor Bush has three judicial openings to fill: one in Pinellas county, one in Broward county, and one in Volusia county. The Judicial Nominating Commission has forwarded him the names of 5 Pinellas county lawyers, 8 Broward county lawyers, and 3 Volusia county lawyers from which to fill the vacancies. In how many different ways can the governor name the three judges?

8. The School Board has to choose two scholarship recipients from among 11 finalists. One senior will get a $40,000 4-year scholarship and another senior will receive a $10,000 one-year scholarship. In how many different ways can the winners be chosen?

9. The School Board has to choose two scholarship recipients from among 11 finalists. Both scholarships are worth $25,000. In how many different ways can the winners be chosen?

10. An election uses the method of sequential pairwise voting. How many different agendas are possible is the number of alternatives is:

a) 5? b) 7?

11. To see if an election has a Condorcet winner, we could look at every possible head-to-head match up. How many such match ups are there if the number of alternatives is:

a) 10? b) 20?

12) The United Nations Security Council consists of five permanent members and 10 non-permanent members. How many different ways can a sub-committee of 9 countries be chosen so that:

a) 4 are permanent members and 5 are non-permanent members?

b) 5 are permanent members and 4 are non-permanent members?

13) A committee consists of 6 Republicans and 4 Democrats. In how many different ways can a sub-committee be formed consisting of 3 Republicans and two Democrats?

14) A senator plans to introduce 8 bills, one of which is an education bill. In how many different orders can he introduce the bills if the education bill must be introduced:

a) first?

b) last?

c) sixth?

15) Six candidates (A, B, C, D, E, and F) are on a ballot in which voters will rank them to obtain a preference list. How many possible preference lists are there that rank candidate C third?

Answers:

1a) 120 b) 1 c) 336 d) 930

2) 720

3) 64

4) 6

5) 462

6) 24

7) 120

8) 110

9) 55

10a) 60 b) 2520

11a) 45 b) 190

12a) 1260 b) 210

13) 120

14a) 5040 b) 5040 c) 5040

15) 120