Section PC.7 Practice Problems

1. In how many ways can 4 different pizza toppings be chosen from 12 available toppings?

2. At a baby shower 17 guests are in attendance and 5 of them are randomly selected to receive a door prize. If all 5 prizes are identical, in how many ways can the prizes be awarded? (Assume no one can receive more than one prize)

3. In the 6/50 lottery game, a player picks six numbers from 1 to 50. How many different choices does the player have if order doesn’t matter?

4. In a lottery daily game, a player picks three numbers from 0 to 9.  How many different choices does the player have if order doesn’t matter?

5. How many ways can 5 different cards be dealt from a standard 52-card deck?

6. A jury pool consists of 18 people. How many different ways can 3 people be chosen to serve on a jury?

7. A jury pool consists of 27 people. How many different ways can 11 people be chosen to serve on a jury and one additional person be chosen to serve as the jury foreman?

8. The United States Senate Committee on Commerce, Science, and Transportation consists of 23 members, 12 Republicans and 11 Democrats. The Surface Transportation and Merchant Marine Subcommittee (STMMS) consist of 8 Republicans and 7 Democrats. How many ways can members of the STMMS be chosen from the Committee?

9. On an exam, a student must pick 2 essay questions from 6 essay questions and 10 multiple choice questions from 20 multiple choice questions to answer. How many different ways can the student pick questions to answer?

10. During the tryouts for a university pep band, there were 4 trumpet players, 12 drummers, and 7 saxophonists. How many ways can the jazz band be chosen so there are 2 trumpet players, 5 drummers, and 3 saxophonists?

11. For each scenario below, would permutations or combinations be appropriate to use?

a. If we want to select r items from n items, and the order of the arrangement is not important, then __________ are used.

b. If we want to select r items from n items, and the order of the arrangement is important, then __________ are used.

c. A medical researcher needs 6 people to test the effectiveness of an experimental drug.  If 13 people have volunteered for the test, in how many ways can 6 people be selected?

d. Fifty people purchase raffle tickets.  Three winning tickets are selected at random.  If first prize is $10000, second prize is 4500, and third prize is $100, in how many different ways can the prizes be awarded?

e. How many different four-letter passwords can be formed from the letters A, B, C, D, E, F, and G if no repetition of letters is allowed?

f. Fifty people purchase raffle tickets.  Here winning tickets are selected at random. If each prize is $400, in how many different ways can the prizes be awarded?

g. In December 2011, the U.S. Senate consisted of 51 Democrats, 47 Republicans, and 2 Independents.  How many distinct five-person committees can be formed if each committee must have 3 Democrats and 2 Republicans?

h. How many ways can a president and a vice president be determined in a club with twelve members?

12. For each scenario below, would permutations or combinations be appropriate to use?

a. Problems involving situations in which the order of the items does not matter.

b. Problems involving situations in which the order of the items makes a difference.

c. How many ways can two people be selected to be co-captains on a team with twelve players?

d. How many ways can a teacher give five different prizes to five of her 25 students?

e. How many ways can a teacher give five identical prizes to five of her 25 students?

f. On an exam, a student must pick 2 essay questions from 6 essay questions and 10 multiple choice questions from 20 multiple choice questions to answer. How many different ways can the student pick questions to answer?

g. A new investor is buying stocks and bonds. An investment company offers 100 stock choices and 85 bond choices.  In how many ways can the investor select 5 stocks and 4 bonds?

h. In how many different ways can 92 different cars be arranged next to one another?

Mixed Practice

1. How many different ways can you display 13 books on a shelf if there are 3 identical math books, 7 identical science books, and 3 identical history books?

2. A high school principal needs to schedule five different classes in five different time periods. How many different class schedules are possible?

3. At a baby shower 22 guests are in attendance and 4 of them are randomly selected to receive a door prize. If all 4 prizes are different, in how many ways can the prizes be awarded? (Assume no one can receive more than one prize)

4. A jury pool consists of 19 people. How many different ways can 3 people be chosen to serve on a jury?

5. A signal can be sent from one location to another by running different colored flags up a flagpole, one above the other. There are 9 different colored flags to choose from. Find the number of different signals consisting of 9 flags, if the middle flag must be brown, the ends must be pink or purple.

6. A gate has a standard key pad with the digits 0 through 9. How many possible code combinations are there if the code is 7 digits long. (Assume repetitions of numbers are allowed and the first digit cannot be a zero)

7. At a baby shower 28 guests are in attendance and 5 of them are randomly selected to receive a door prize. If all 5 prizes are identical, in how many ways can the prizes be awarded? (Assume no one can receive more than one prize)

8. On an exam, a student must pick 3 essay questions from 6 essay questions and 7 multiple choice questions from 17 multiple choice questions to answer. How many different ways can the student pick questions to answer?

9. The United States Senate Committee on Commerce, Science, and Transportation consists of 21 members, 12 Republicans and 9 Democrats. The Surface Transportation and Merchant Marine Subcommittee (STMMS) consists of 8 Republicans and 5 Democrats. How many ways can members of the STMMS be chosen from the Committee?

10. A computer password must be 6 characters long. How many passwords are possible if only the 26 lowercase letters of the alphabet are allowed? (Assume repetitions of letters are allowed)

11. A jury pool consists of 15 people. How many different ways can 3 people be chosen to serve on a jury and one additional person be chosen to serve as the jury foreman?

12. Bailey has 5 skirts, 8 blouses, and 8 pairs of shoes. How many different skirt-blouse-shoe outfits can she wear?