Name Algebra 1B notes and problems April 4, 2008 Counting 3: Combinations page 1

Counting methods (Part 3): Combinations From before: permutation A permutation problem is a problem that involves counting the number of ways that some of a set of things can be selected with an order. To count the number of ways that r out of n things can be selected with an order, multiply r whole numbers starting from n and counting downward. The answer to this problem is called “n permutation r” and abbreviated nPr .

New problem type: combination A combination problem is a problem that involves counting the number of ways that some things can be selected without choosing an order. Here’s an example: Example: There are 15 tracks on a CD. You are asked to choose your 5 favorite tracks, but you don’t have to specify an order. How many ways can this be done? (15Ч14 Ч13Ч12 Ч11) To get the answer: Calculate . The answer is 3,003. (1Ч2 Ч3Ч4 Ч5) Another example: Example: A club with 25 members wishes to select a 4-member leadership team. How many ways can this be done? (25Ч24 Ч23Ч22) To get the answer: Calculate . The answer is 12,650. (1Ч2 Ч3Ч4) Here is a general statement of the combination counting method: To count the number of ways that r out of n things can be selected without an order, set 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. The answer to this problem is called “n combination r” and abbreviated nCr . Note the symbol that is used to denote the answer to a combination problem. The answers to the example problems above are called 15C5 and 25C4.

You try it: combinations 1. Find the values of these combination number symbols.

a. 5C3

b. 50C2 Name Algebra 1B notes and problems April 4, 2008 Counting 3: Combinations page 2

Directions: For each counting problem, first identify the answer as a combination number, then write the answer as a fraction, and finally find the answer as a plain number. (15Ч14 Ч13Ч12 Ч11) [Example of an answer: 15C5 = = 3003.] (1Ч2 Ч3Ч4 Ч5) 2. From a box of 16 color crayons, a child selects a set of 4 crayons as her favorites. How many different ways can she make this selection?

3. There are 35 students on a football team. They must elect 3 students to serve as captains. How many different ways could the election turn out?

4. A store has to hire two cashiers. Five people are interviewed for the jobs. How many different ways can the hiring decisions be made?

5. Suppose that your English teacher asks you to choose 2 of these 4 books to read. Beloved Hamlet Joy Luck Club To Kill a Mockingbird How many different ways could you choose?

(8Ч7Ч6) 6. Make up your own counting problem where the answer would be 8C3 = = 56. (1Ч2Ч3) (Think of your own new counting situation; don’t just borrow from a problem you’ve already seen.) Name Algebra 1B notes and problems April 4, 2008 Counting 3: Combinations page 3

Combination operation on your calculator Your calculator has a shortcut operation that gives the answers to combination problems without having to multiply and divide.

For example, the answer to problem 4 is called 5C2. Here is how to find 5C2 on your calculator. Keys to type: What you’ll see on the screen: 5 MATH <- 3 2 ENTER

You try it: Combination operation on your calculator Directions: Find the answers to these counting problems using the calculator’s combination operation as shown at the top of this page. First identify the combination needed (example: 5C2) then write the answer (example: 10). 7. From a box of 16 color crayons, a child selects a set of 4 crayons as her favorites. How many different ways can she make this selection?

8. You go to the video store to rent some movies for the weekend. There are 1000 movies available, and you want to select 3 of them. How many ways can you make your choices?

Something different happens in this problem: 9. An essay contest is announced, with an award to be given to each of five winning entries. Suppose that only five essays are submitted to the contest. How many ways can the awards be given? Name Algebra 1B notes and problems April 4, 2008 Counting 3: Combinations page 4

Permutation and combination problems together When you encounter a problem that asks you to count the number of ways that a group can be selected from a larger group, you need to figure out whether it is a permutation problem or a combination problem. Here’s how to decide.  If the problem situation involves the selections being made in a particular order or for specific positions, it’s a permutation counting problem (nPr).  If the order in which the selections are made doesn’t matter in the problem situation, it’s a combination counting problem (nCr). Think carefully about this distinction as you do the following problems.

You try it: permutations and combinations 10. From a class with 20 students, I need to select a group of 9 students to perform a top-secret mission. How many different ways can this group be selected?

a. Which of these is the correct calculation: 20P9 or 20C9 ?

b. Now use your calculator to get the answer.

11. From a class of with 20 students, I need to select a group of 9 students to play softball. This means that besides choosing the students, I need to make a batting order (decide who will bat 1st, 2nd, 3rd, etc.). How many different ways can this be done?

a. Which of these is the correct calculation: 20P9 or 20C9 ?

b. Now use your calculator to get the answer.

12. A hiker would like to invite 7 friends to go on a trip, but has room for only 4 of them. In how many different ways can they be chosen?

13. The hiker with 7 friends wants to make a list of her 1st, 2nd, 3rd, and 4th best friends. How many different possible lists are there? Name Algebra 1B notes and problems April 4, 2008 Counting 3: Combinations page 5

14. An ice cream stand carries 5 flavors: vanilla, chocolate, strawberry, v c v orange, and pineapple. The stand sells two-scoop ice cream cones. Some examples are shown in the picture. The two flavors chosen cannot c v s be the same. Note that vanilla-chocolate and chocolate-vanilla are considered to be different cones. In other words, the order of the scoops matters. a. How many different kinds of two-scoop ice cream cones are there?

b. List all of the possible two-scoop ice cream cones. You can abbreviate just using the first letter of each flavor.

15. The same ice cream stand also sells two-scoop ice cream dishes. Some examples are shown in the picture. The two flavors chosen v c v s cannot be the same. In a dish, vanilla-chocolate would be considered the same dish as chocolate-vanilla, so you shouldn’t list them both. a. How many different kinds of two-scoop ice cream dishes are there?

b. List all of the possible two-scoop ice cream dishes. You can abbreviate just using the first letter of each flavor.

16. The answer to the ice cream dish problem (15a) was half of the answer to the ice cream cone problem (14a). Explain why this happened. Name Algebra 1B notes and problems April 4, 2008 Counting 3: Combinations page 6

17. The ice cream stand has made some changes. There are now 7 flavors instead of 5, and it is additionally selling 3-scoop dishes and cones. a. With 7 flavors, how many 2-scoop dishes are there?

b. With 7 flavors, how many 2-scoop cones are there?

c. With 7 flavors, how many 3-scoop dishes are there?

d. With 7 flavors, how many 3-scoop cones are there?

Think of your own new counting situations for the next two questions; don’t just borrow situations from problems you’ve already seen.

18. Make up your own problem where the answer would be found using 12P5.

19. Make up your own problem where the answer would be found using 12C5.