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10-7 Combinations Warm Up Problem of the Day Lesson Presentation Lesson Quizzes 10-7 Combinations Warm Up Decide 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. Dependent; there are fewer cards from which to choose. 2. the chance of it raining on the third Tuesday of the month after it has rained on the first Tuesday 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 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? 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. 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 10-7 Combinations Additional Example 1 Continued 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. 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 dishes Kareem 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. 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 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. 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 There are 4 possible combinations of 3 shirts that Jim can choose to bring on his trip. 10-7 Combinations Additional Example 2: Problem Solving Application 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? 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. 10-7 Combinations Additional Example 2 Continued 2 Make a Plan You can make a tree diagram to show the possible combinations. 10-7 Combinations Additional Example 2 Continued 3 Solve Strawberry Vanilla Vanilla Chocolate Chocolate Strawberry Chocolate Strawberry 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 Additional Example 2 Continued 4 Look Back 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 2 + 1 = 3. 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? 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. 10-7 Combinations Check It Out: Example 2 Continued 2 Make a Plan You can make a tree diagram to show the possible combinations. 10-7 Combinations Check It Out: Example 2 Continued 3 Solve Sugar Oatmeal Chocolate chip Sugar Chocolate chip Oatmeal Oatmeal Chocolate chip Sugar The tree diagram shows 6 possible ways to combine two types, but each combination is listed twice. So there are 6 ÷ 2 = 3 possible combinations. 10-7 Combinations Check It Out: Example 2 Continued 4 Look Back 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 2 + 1 = 3. 10-7 Combinations Lesson Quizzes Standard Lesson Quiz 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? 21 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? 6 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 10-7 Combinations Lesson Quiz for Student Response Systems 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 10-7 Combinations Lesson Quiz for Student Response Systems 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 10-7 Combinations Lesson Quiz for Student Response Systems 3. How many 2-letter combinations are possible from the letters in the word MATH? A. 2 B. 4 C. 6 D. 8