Name: Algebra 2

Date:

Counting Problems: Practice

First, here’s a review of the counting methods we’ve studied, and some advice on which method tochoose. Your assignment begins on page 2. Homework is to finish whatever you don’t finish in class.

Once again, the four counting methods

·  Multiplication Principle: When you have two or more decisions to make, the total number of possible combinations is found by multiplying the numbers of options for each decision.

·  Factorial (number of ways to put in order): If you have n objects that need to be arranged in anorder, the number of possible orders is “n factorial,” abbreviated n! .

□  You can find n! by multiplying all the whole numbers from n down to 1.

·  Permutations (number of ways to select with an order): If you are asked to count the number of ways that routof n objects can be selected with an order, the number of ways is“n permutation r,” abbreviated nPr .

□  You can find nPr by multiplying r whole numbers starting from n and counting downward.

·  Combinations (number of ways to select without an order): If you are asked to count the number of ways that routof n objects can be selected without an order, the number of ways is “ncombination r,” abbreviated nCr .

□  You can find nCr by setting up a fraction in the following way:
on the top, multiply r whole numbers counting downward from n;
on the bottom, multiply r whole numbers counting upward from 1.

Choosing which method to use

·  If the problem involves making two or more separate decisions and asks about the combined number of possibilities, use the Multiplication Principle.

·  If the problem involves a group of objects, and asks a question about how many ways they can be selected or assigned or put in order, use the flowchart below.

Review problems

Directions for problems 1 through 9: Answer these counting questions. For now, do not use the shortcut operations on the calculator. It’s OK to use your calculator just for arithmetic. Here’s an example of the amount of work you must show: = 35.

1. Hannah is signing up for a summer day camp. She has to pick 4 out of the 9 activities offered by the camp. How many different ways can she make her choices?

2. The principal has to decide the schedule for a half-day. That is, she needs to pick four of the eight class letter blocks, with an order. (For example, the choice could be “C, B,G, F.”) Howmany different schedules are possible?

3. Carly and Jake went to an arcade with 8 different games.

a. Carly decides she wants to play each of the games once. In how many different orders could she decide to play the games?

b. Jake only has enough tokens to play 5 out of the 8 games. In how many different orders could he decide to play the games?

4. The computer in a library children’s room has a password that is just two characters long. Each character can be a capital letter or a number. Here are some examples of possible passwords: Z9, QW, 37, 4T, KK. How many different passwords are possible?

5. The Drama Club has 25 members. It elects 4 different members as president, vicepresident, a secretary, and a treasurer. How many different ways can this be done?

6. The Karate Club has 25 members. It elects 4 of its members as a leadership team. Howmany different ways can this be done?

7. A survey asks respondents to rate each of these 5 cereals (Cheerios, Life, Apple Jacks, Frosted Flakes, Crispix) on a 1-to-10 scale. Which of these is the number of different ways the survey could be completed: 510 or 105 ? Decide which, then find the value.

8. There are 20 citizens available to serve on a jury. Of them, 12 must be chosen to form a jury. How many ways can the jury be formed?

9. There are 24 basketball teams that could play in a tournament. From them, 16 teams must be chosen for the tournament, with each chosen team given a “seed” of 1st, 2nd, 3rd, …, 16th.
How many different ways can the tournament be formed?


10. There are 5 members on the Portland City Council. The council members need to choose a 3-person subcommittee.

a. How many different possibilities are there for the members of a 3-person subcommittee?

b. The Mayor of Portland is one of the 5 members of the City Council. How many different possibilities are there for the members of a 3-person subcommittee,
if the Mayor must be included as one of the subcommittee members?
Hint: The subcommittee will consist of the Mayor plus 2 of the other 4.

c. The Mayor of Portland is one of the 5 members of the City Council. How many different possibilities are there for the members of a 3-person subcommittee,
if the Mayor must not be included as one of the subcommittee members?

11. A sandwich restaurant offers 9 types of sandwich, 5 types of bread, 3 types of chips, and 6types of drink. How many different meal choices (of sandwich, bread, chips, and drink) arethere?

12. A different restaurant offers a “You Pick Two” deal. There are 8 types of sandwich, 5 types of soup, and 6 types of salad. A meal consists of choices from two different categories.

a. How many different ways are there to order a sandwich and a soup?

b. How many different ways are there to order a sandwich and a salad?

c. How many different ways are there to order a soup and a salad?

d. How many different meals are possible in all?
Hint: Combine your answers to parts a, b, and c.

13. An LCD digit display (used in electronic devices such as digital clocks) has seven segments, eachof which can be turned on or off. For example, todisplay the number 2, five of the segments are turned on and the other two are turned off, as shown in the picture.

Think about all the different possible appearances of this display (not just the ways that represent numbers). How many different displays are possible?

14. A club has 16 members.

a. In how many different ways could the club choose a president and a vice-president?

b. In how many different ways could the club choose two co-presidents?

c. In how many different ways could the club choose a president, vice-president, and secretary?

d. In how many different ways could the club choose a three-member governing committee?

e. In how many different ways could the club choose two co-presidents and a secretary?

15. An album has 15 songs. How many ways can you select your favorite, second favorite, and third favorite songs?

16. A class of 25 students needs to select a group of 4 students as their representatives. How many ways can this be done?

17. You flip a coin 8 times. How many total outcomes are possible?

Some ANSWERS

1. 9C4 = = 126.

2. 8P4 = 8 · 7· 6 · 5 = 1680.

3. a. 8! = 8 · 7 · 6 · 5 · 4 · 3 · 2 · 1 = 40,320.

b. 8P5 = 8 · 7 · 6 · 5 · 4 = 6,720.

4. 36 · 36 = 1,296.

5. 25P4 = 25 · 24 · 23 · 22 = 303,600.

6. 25C4 = = 12,650.

7. 105 = 100,000.

8. 20C12 = 125,970.

9. 24P16 ≈ 15,388,105,000,000,000,000.

10. a. 5C3 =

b. 4C2 =

c. 4C3 =

11. 9 · 5 · 3 · 6 = 810.

12. a. 8 · 5 = 40 b. 8 · 6 = 48 c. 5 · 6 = 30 d. 40 + 48 + 30 = 118.

13. 27 = 128.

14. a. 16P2 = 240 b. 16C2 = 120 c. 16P3 = 3360 d. 16C3 = 560

e. 16C2∙14 = 1680 or 16∙ 15C2 = 1680

15. 15P3 = 2730

16. 25C4 = 12,650 17. 28 = 256