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THE ICE CREAM SOCIAL

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Choose one flavor of ice cream Chocolate Vanilla Strawberry CHOOSE TW0 TOPPINGS: CHOCOLATE SYRUP WHIPPED CREAM NUTS CHERRIES PINEAPPLE

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THE ICE CREAM SOCIAL Rewarded with an ice cream social, students may select one flavor of ice cream and two toppings. There are three flavors of ice cream and five toppings to choose from. How many combinations of ice cream and toppings are possible? What is the probability that you will correctly guess the combination a student will order?

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TO VISUALIZE THE SAMPLE SPACE YOU CAN CREATE A TREE DIAGRAM ChocolateVanilla Strawberry Choose a flavor of ice cream

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TO VISUALIZE THE SAMPLE SPACE YOU CAN CREATE A TREE DIAGRAM Chocolate Chocolate Syrup Whipped Cream Nuts Cherries Pineapple Vanilla Chocolate Syrup Whipped Cream Nuts Cherries Pineapple Strawberry Chocolate Syrup Whipped Cream Nuts Cherries Pineapple Choose the first topping

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TO VISUALIZE THE SAMPLE SPACE YOU CAN CREATE A TREE DIAGRAM Chocolate Syrup Whipped Cream Nuts Cherries Pineapple Whipped Cream Nuts Cherries Pineapple Nuts Cherries Pineapple Cherries Pineapples Pineapple No more unique combinations Choose the second topping Chocolate 4 choices 3 choices 2 choices 1 choices 0 choices 10 Total choices

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TO VISUALIZE THE SAMPLE SPACE YOU CAN CREATE A TREE DIAGRAM Chocolate Syrup Whipped Cream Nuts Cherries Pineapple Whipped Cream Nuts Cherries Pineapple Nuts Cherries Pineapple Cherries Pineapples Pineapple No more unique combinations The same choices can be made for vanilla Vanilla

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TO VISUALIZE THE SAMPLE SPACE YOU CAN CREATE A TREE DIAGRAM Chocolate Syrup Whipped Cream Nuts Cherries Pineapple Whipped Cream Nuts Cherries Pineapple Nuts Cherries Pineapple Cherries Pineapples Pineapple No more unique combinations. and for strawberry, for a total of 3 x 10 or 30 choices. Strawberry

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DETERMINE THE SAMPLE SPACE If we could use the multiplication rule for this problem, we would have. 3 Ice cream flavor s x 5 choices for the 1st topping x 4 choices for the 2 nd topping Total possible combinations is 3 x 5 x 4 = 60 However, the tree diagram reveals there are only 30 unique choices. The probability of guessing the order correctly is 1/30. This problem involves the MULTIPLICATION RULE and a COMBINATION. The number of combinations of 3 ice cream flavors and 5 toppings, taken 2 at a time, is 3 x 5 C 2 = 3 x 5! = 3 x 5!___ = 3 x 5 x 4 x 3 x 2 x 1 = 30 2!(5-2)! 2!(3)! 2 x 1 x 3 x 2 x 1

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MULTIPLICATION RULE AND COMBINATIONS Suppose students may choose a third topping. What is the size of the new sample space.? Rather than completing the tree, you can use the combination formula to calculate the size of the sample space. 3 Ice cream flavor s x 5 toppings taken 3 at a time. Total possible combinations is 30 The number of combinations of 3 ice cream flavors and 5 toppings, taken 3 at a time, is 3 x 5 C 3 = 3 x 5! = 3 x 5!___ = 3 x 5 x 4 x 3 x 2 x 1 = 30 3!(5-3)! 3!(2)! 3 x 2 x 1 x 2 x 1

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EXTENSION PROBLEM Marble Slab is famous for its flavors and toppings. How many combinations are possible if the High School moves its ice cream social to Marble Slab. To calculate the number of samples, you need to know the number of flavors and the number of toppings. Visit the Marble Slab site to find out. Assume all the flavors and toppings are available.Marble Slab

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MARBLE SLAB There are 67 ice cream flavors and 68 toppings. If all the flavors and toppings are available, The number of combinations of 67 ice cream flavors and 68 toppings, taken 2 at a time, is 67 x 68 C 2 = ? A problem of this magnitudes calls for a little help from technology.

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