1. 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. 2. the chance of it raining on the third Tuesday of the month after it has rained on the first Tuesday Course 2 10-6 Combinations

2. Learn to find the number of possible combinations. Course 2 10-6 Combinations

3. Vocabulary combination Insert Lesson Title Here Course 2 10-6 Combinations

4. 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 2 10-6 Combinations

5. A combination is a grouping of objects or events in which the order does not matter. For example, the letters A, B, and C can be arranged in 6 different ways: ABC, ACB, BCA, BAC, CAB, and CBA. Since the order does not matter, each arrangement represents the same combination. One way to find possible combinations is to make a list. Course 2 10-6 Combinations

6. Additional Example 1: Using a List to Find Combinations Kristy’s Diner offers customers a choice of 4 side dishes with each order: carrots, corn, french fries, and mashed potatoes. In how many different ways can Kareem choose 3 side dishes with his order? Begin by listing 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 2 10-6 Combinations

7. 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 2 10-6 Combinations

8. Try This: 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. In how many different ways can he choose the 3 shirts for his trip? Begin by listing 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 2 10-6 Combinations

9. Try This: 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 2 10-6 Combinations

10. 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 2 10-6 Combinations

11. Additional Example 2 Continued 2 Make a Plan You can make a tree diagram to show the possible combinations. Course 2 10-6 Combinations

12. 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 2 10-6 Combinations

13. Look Back 4 You can also check by making a list. The vanilla can be paired with two other flavors and the chocolate with one. The total number of possible pairs is 2 + 1 = 3. Additional Example 2 Continued Course 2 10-6 Combinations

14. Try This: 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 2 10-6 Combinations

15. Try This: Example 2 Continued Insert Lesson Title Here 2 Make a Plan You can make a tree diagram to show the possible combinations. Course 2 10-6 Combinations

16. Try This: 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 2 10-6 Combinations

17. Try This: Example 2 Continued Insert Lesson Title Here Look Back 4 You can also check by making a list. 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. Course 2 10-6 Combinations

18. 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. A salad bar offers carrots, cheese, croutons, hard-boiled eggs, onions, and tomatoes as salad toppings. How many combinations of 3 toppings can Jamie put on her salad? 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? Insert Lesson Title Here Course 2 10-6 Combinations

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