11-6 Combinations

Course 2

Warm UpWarm Up

Problem of the DayProblem of the Day

Lesson PresentationLesson Presentation

Learn to find the number of possible combinations.

Course 2

11-6 Combinations

Vocabulary

combination

Insert Lesson Title Here

Course 2

11-6 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?

Course 2

11-6 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.

Course 2

11-6 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.

1, 2, 3 2, 1, 3 3, 1, 2 4, 1, 2

1, 2, 4 2, 1, 4 3, 1, 4 4, 1, 3

1, 3, 4 2, 3, 4 3, 2, 4 4, 2, 3Course 2

11-6 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

1, 2, 3 2, 1, 3 3, 1, 2 4, 1, 2

1, 2, 4 2, 1, 4 3, 1, 4 4, 1, 3

1, 3, 4 2, 3, 4 3, 2, 4 4, 2, 3

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

Course 2

11-6 Combinations

Check It Out: Example 1

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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.

r, g, b g, r, b b, r, g y, r, g

r, g, y g, r, y b, r, y y, r, b

r, b, y g, b, y b, g, y y, g, b

Course 2

11-6 Combinations

Check It Out: Example 1 Continued

Insert Lesson Title Here

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.

r, g, b g, r, b b, r, g y, r, g

r, g, y g, r, y b, r, y y, r, b

r, b, y g, b, y b, g, y y, g, b

Course 2

11-6 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.

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11-6 Combinations

Additional Example 2 Continued

22 Make a Plan

You can make a tree diagram to show thepossible combinations.

Course 2

11-6 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.

Course 2

11-6 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

Course 2

11-6 Combinations

Check It Out: Example 2

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Hamilton had three flavors 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.

Course 2

11-6 Combinations

Check It Out: Example 2 Continued

Insert Lesson Title Here

22 Make a Plan

You can make a tree diagram to show thepossible combinations.

Course 2

11-6 Combinations

Check It Out: Example 2 Continued

Insert Lesson Title Here

Solve33

OatmealChocolate chip

SugarSugar

Chocolate chip

Oatmeal

Chocolate chipOatmeal

Sugar

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

Course 2

11-6 Combinations

Check It Out: Example 2 Continued

Insert Lesson Title Here

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.

Course 2

11-6 Combinations