11-6 Combinations Course 2 Warm Up Warm Up Problem of the Day Problem of the Day Lesson Presentation Lesson Presentation

Learn to find the number of possible combinations. Course Combinations

Vocabulary combination Insert Lesson Title Here Course Combinations

Mrs. Logan’s class has to read any two of the following books. 1. The Adventures of Tom Sawyer, by Mark Twain 2. The Call of the Wild, by Jack London 3. 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 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 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, 32, 1, 33, 1, 24, 1, 2 1, 2, 42, 1, 43, 1, 44, 1, 3 1, 3, 42, 3, 43, 2, 44, 2, 3 Course 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, 32, 1, 33, 1, 24, 1, 2 1, 2, 42, 1, 43, 1, 44, 1, 3 1, 3, 42, 3, 43, 2, 44, 2, 3 There are 4 possible combinations of 3 side dishes Kareem can choose with his order. Course Combinations

Check It Out: Example 1 Insert Lesson Title Here 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, bg, r, bb, r, gy, r, g r, g, yg, r, yb, r, yy, r, b r, b, yg, b, yb, g, yy, g, b Course 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 shirts that Jim can choose to bring on his trip. r, g, bg, r, bb, r, gy, r, g r, g, yg, r, yb, r, yy, r, b r, b, yg, b, yb, g, yy, g, b Course 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 1 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. Course Combinations

Additional Example 2 Continued 2 Make a Plan You can make a tree diagram to show the possible combinations. Course Combinations

Additional Example 2 Continued Solve 3 Vanilla Chocolate Strawberry Chocolate Vanilla Strawberry Chocolate 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 Combinations

Look Back 4 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 possible pairs is = 3. Additional Example 2 Continued Course Combinations

Check It Out: Example 2 Insert Lesson Title Here 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? 1 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 Combinations

Check It Out: Example 2 Continued Insert Lesson Title Here 2 Make a Plan You can make a tree diagram to show the possible combinations. Course Combinations

Check It Out: Example 2 Continued Insert Lesson Title Here Solve 3 Oatmeal Chocolate chip Sugar Chocolate chip Oatmeal Chocolate chip Oatmeal Sugar 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 Combinations

Check It Out: Example 2 Continued Insert Lesson Title Here Look Back 4 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 possible pairs is = 3. Course Combinations