10-7 Combinations

Warm UpWarm Up

Lesson PresentationLesson Presentation

Problem of the DayProblem of the Day

Lesson QuizzesLesson Quizzes

10-7 Combinations

Warm UpDecide whether each event is independent or dependent. Explain your answer.

1. Bill picks a king from a pile of cards and keeps it. On his next turn he tries for a queen.

2. the chance of it raining on the third Tuesday of the month after it has rained on the first Tuesday

Dependent; there are fewer cards from which to choose.

Independent; the rain on the first Tuesday has no effect on the weather two weeks later.

10-7 Combinations

Problem of the Day

There are 7 players. Can more teams of 2 different people or 5 different people be formed? (Hint: Look for a shortcut.)

The same number of both can be formed. For every 5-person team there is a 2-person team (the 2 players left out).

10-7 Combinations

I can find the number of possible combinations.

10-7 Combinations

Vocabulary

combination

10-7 Combinations

Mrs. Logan’s class has to read any two of the following books.

1. The Adventures of Tom Sawyer, by Mark Twain2. The Call of the Wild, by Jack London3. A Christmas Carol, by Charles Dickens

4. Treasure Island, by Robert Louis Stevenson

5. Tuck Everlasting, by Natalie Babbit

How many possible combinations of books could the students choose?

10-7 Combinations

A combination is a grouping of objects or events in which the order does not matter. For example, a student can choose books 1 and 2 or books 2 and 1. Since the order does not matter, the two arrangements represent the same combination. One way to find possible combinations is to make a table.

10-7 Combinations

Additional Example 1: Using a Table to Find Combinations

Kristy’s Diner offers customers a choice of 4 side dishes with each order: carrots, corn, french fries, and mashed potatoes. How many different combinations of 3 side dishes can Kareem choose?

Begin by making a table showing all the possible choices of side dishes taken three at a time.

4, 2, 33, 2, 42, 3, 41, 3, 4

4, 1, 33, 1, 42, 1, 41, 2, 4

4, 1, 23, 1, 22, 1, 31, 2, 3

10-7 Combinations

Because order does not matter, you can eliminate repeated triples. For example 1, 2, 3 is already listed, so 2, 1, 3 can be eliminated.

Additional Example 1 Continued

4, 2, 33, 2, 42, 3, 41, 3, 4

4, 1, 33, 1, 42, 1, 41, 2, 4

4, 1, 23, 1, 22, 1, 31, 2, 3

There are 4 possible combinations of 3 side dishesKareem can choose with his order.

10-7 Combinations

Check It Out: Example 1

Jim is packing for a trip. He has 4 different colors of shirts: red, green, blue, and yellow(r, g, b, y). He only has room for 3 of them. How many different combinations of 3 shirts can Jim choose for his trip?

Begin by making a table of all the possible choices of shirts taken three at a time.

y, g, bb, g, yg, b, yr, b, y

y, r, bb, r, yg, r, yr, g, y

y, r, gb, r, gg, r, br, g, b

10-7 Combinations

Check It Out: Example 1 Continued

Because order does not matter, you can eliminate repeated triples. For example r, g, b is already listed so g, r, b can be eliminated.

There are 4 possible combinations of 3 shirtsthat Jim can choose to bring on his trip.

y, g, bb, g, yg, b, yr, b, y

y, r, bb, r, yg, r, yr, g, y

y, r, gb, r, gg, r, br, g, b

10-7 Combinations

Lara is going to make a double-dip cone from a choice of vanilla, chocolate, and strawberry. She wants each dip to be a different flavor. How many different cone combinations can she choose from?

Additional Example 2: Problem Solving Application

11 Understand the Problem

Rewrite the question as a statement.• Find the number of possible combinations of

two flavors Lara can choose.

List the important information:

• There are three flavor choices in all.

10-7 Combinations

Additional Example 2 Continued

22 Make a Plan

You can make a tree diagram to show thepossible combinations.

10-7 Combinations

Additional Example 2 Continued

Solve33

VanillaChocolate

StrawberryChocolate

Vanilla

Strawberry

StrawberryChocolate

Vanilla

The tree diagram shows 6 possible ways to combine two flavors, but each combination is listed twice. So there are 6 ÷ 2 = 3 possible combinations.

10-7 Combinations

Look Back44

You can also check by making a table. The vanilla can be paired with two other flavors and the chocolate with one. The total number of possiblepairs is 2 + 1 = 3.

Additional Example 2 Continued

10-7 Combinations

Check It Out: Example 2

Hamilton had three types of cookies: sugar, chocolate chip, and oatmeal. He wants to put them into packages, with each package containing two different flavors of cookies. How many different packages can he make?

11 Understand the Problem

Rewrite the question as a statement.

• Find the number of possible combinations that Hamilton can choose.List the important information:

• There are three cookie types in all.

10-7 Combinations

Check It Out: Example 2 Continued

22 Make a Plan

You can make a tree diagram to show thepossible combinations.

10-7 Combinations

Check It Out: Example 2 Continued

Solve33

OatmealChocolate chip

SugarSugar

Chocolate chip

Oatmeal

Chocolate chipOatmeal

Sugar

The tree diagram shows 6 possible ways to combinetwo types, but each combination is listed twice. So there are 6 ÷ 2 = 3 possible combinations.

10-7 Combinations

Check It Out: Example 2 Continued

Look Back44

You can also check by making a table. The oatmeal can be combined with two other types and the chocolate chip with one. The total number of possiblepairs is 2 + 1 = 3.

10-7 Combinations

Standard Lesson Quiz

Lesson Quizzes

Lesson Quiz for Student Response Systems

10-7 Combinations

Lesson Quiz

1. A pizzeria has on special a pizza with 2 toppings for $9.95. It offers pepperoni, olives, tomato, sausage, bacon, mushrooms, and ham as toppings. How many different specials could you build?

2. Jaime has a red shirt and a blue shirt. She has a green scarf, a yellow scarf, and a purple scarf. How many different shirt-scarf combinations can she wear?

3. Karl has to pick two reptiles to write a report on. If there are 8 possible reptiles to choose from, how many pairs are possible? 28

21

6

10-7 Combinations

1. An ice cream parlor offers a discount on a triple sundae that is made of 3 flavors. You can choose from chocolate, vanilla, strawberry, mint, and butterscotch. How many different combinations are possible? A. 5

B. 6

C. 8

D. 10

Lesson Quiz for Student Response Systems

10-7 Combinations

2. On the opening day, a restaurant offers free drinks and desserts to all the customers. A customer can choose from apple juice, orange juice, grape juice, and cranberry juice. He can choose from ice cream, cake, and pudding as desserts. How many different juice-dessert combinations are possible? A. 7

B. 12

C. 15

D. 24

Lesson Quiz for Student Response Systems

10-7 Combinations

3. How many 2-letter combinations are possible from the letters in the word MATH? A. 2

B. 4

C. 6

D. 8

Lesson Quiz for Student Response Systems