MGF 1107/ Combinatorics/ Homework

1. Simplify:

a) 5! b) 0! c) d) e) Chapter 11, exercise #18abcd (p. 364)

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?
16) A city commissioner has to make his schedule for an upcoming day. He has time for three events, one in the morning, one in the afternoon, and one in the evening. He has decided to visit a school in the morning, a business organization in the afternoon, and a condominium association meeting in the evening. Assuming he has invitations from 6 schools, 3 business organizations, and 4 condominium associations, in how many different ways can the commissioner form his schedule?
17) The Florida Legislature meets once a year for two months in Tallahassee. A freshman legislator has been told by her party leader that, as a new member of the House, she can introduce only four of the 11 bills she had hoped to pass, one every two weeks. The leader informs her that she should introduce the bill that is her biggest priority first since it will have more time to go through the committee process and that the later a bill is introduced in a session, the less chance it has of passing. In how many different ways can she choose the four bills she will introduce?
18) A dinner is held to honor the outgoing mayor for his many years of service. The menu gives each attendee a choice of 4 appetizers, 6 entrees, and 3 desserts. In how many different ways can an attendee choose their three courses?
19) Suppose there are 14 females and 16 males in the Student Government. Four representatives are to be selected to attend the State Conference.
a) How many different ways are there to select a group of four students to attend the conference?
b) If it is decided to send two females and two males, how many different selections are possible?

Answers:

1a) 120 b) 1 c) 336 d) 930 e) 20, 4950, 4950, 126

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
16) 72
17) 7920
18) 72

19a) 27,405 b) 10,920