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Copyright © Cengage Learning. All rights reserved. 6 Sets and Counting

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Copyright © Cengage Learning. All rights reserved. 6.3 Decision Algorithms: The Addition and Multiplication Principles

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3 Let’s start with a really simple example. You walk into an ice cream parlor and find that you can choose between ice cream, of which there are 15 flavors, and frozen yogurt, of which there are 5 flavors. How many different selections can you make? Clearly, you have 15 + 5 = 20 different desserts from which to choose. Mathematically, this is an example of the formula for the cardinality of a disjoint union: If we let A be the set of ice creams you can choose from, and B the set of frozen yogurts, then A ∩ B = ∅ and we want n(A U B).

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4 But the formula for the cardinality of a disjoint union is n(A U B) = n(A) + n(B), which gives 15 + 5 = 20 in this case. This example illustrates a very useful general principle. Addition Principle When choosing among r disjoint alternatives, suppose that alternative 1 has n 1 possible outcomes, alternative 2 has n 2 possible outcomes,... alternative r has n r possible outcomes, with no two of these outcomes the same. Then there are a total of n 1 + n 2 + · · · + n r possible outcomes. Decision Algorithms: The Addition and Multiplication Principles

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5 Quick Example At a restaurant you can choose among 8 chicken dishes, 10 beef dishes, 4 seafood dishes, and 12 vegetarian dishes. This gives a total of 8 + 10 + 4 + 12 = 34 different dishes to choose from. Here is another simple example. In that ice cream parlor, not only can you choose from 15 flavors of ice cream, but you can also choose from 3 different sizes of cone. How many different ice cream cones can you select from? Decision Algorithms: The Addition and Multiplication Principles

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6 This time, we want to choose both a flavor and a size, or, in other words, a pair (flavor, size). Therefore, if we let A again be the set of ice cream flavors and now let C be the set of cone sizes, the pair we want to choose is an element of A × C, the Cartesian product. To find the number of choices we have, we use the formula for the cardinality of a Cartesian product: n(A C) = n(A)n(C). In this case, we get 15 3 = 45 different ice cream cones we can select. Decision Algorithms: The Addition and Multiplication Principles

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7 This example illustrates another general principle. Multiplication Principle When making a sequence of choices with r steps, suppose that step 1 has n 1 possible outcomes step 2 has n 2 possible outcomes... step r has n r possible outcomes and that each sequence of choices results in a distinct outcome. Then there are a total of n 1 n 2 · · · n r possible outcomes. Decision Algorithms: The Addition and Multiplication Principles

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8 Quick Example At a restaurant you can choose among 5 appetizers, 34 main dishes, and 10 desserts. This gives a total of 5 34 10 = 1,700 different meals (each including one appetizer, one main dish, and one dessert) from which you can choose. Decision Algorithms: The Addition and Multiplication Principles

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9 Example 1 – Desserts You walk into an ice cream parlor and find that you can choose between ice cream, of which there are 15 flavors, and frozen yogurt, of which there are 5 flavors. In addition, you can choose among 3 different sizes of cones for your ice cream or 2 different sizes of cups for your yogurt. How many different desserts can you choose from? Solution: It helps to think about a definite procedure for deciding which dessert you will choose. Here is one we can use: Alternative 1: An ice cream cone Step 1 Choose a flavor. Step 2 Choose a size.

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10 Example 1 – Solution Alternative 2: A cup of frozen yogurt Step 1 Choose a flavor. Step 2 Choose a size That is, we can choose between alternative 1 and alternative 2. If we choose alternative 1, we have a sequence of two choices to make: flavor and size. The same is true of alternative 2. cont’d

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11 Example 1 – Solution We shall call a procedure in which we make a sequence of decisions a decision algorithm. Once we have a decision algorithm, we can use the addition and multiplication principles to count the number of possible outcomes. Alternative 1: An ice cream cone Step 1 Choose a flavor; 15 choices Step 2 Choose a size; 3 choices There are 15 3 = 45 possible choices in alternative 1. Multiplication Principle cont’d

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12 Example 1 – Solution Alternative 2: A cup of frozen yogurt Step 1 Choose a flavor; 5 choices Step 2 Choose a size; 2 choices There are 5 2 = 10 possible choices in alternative 2. So, there are 45 + 10 = 55 possible choices of desserts. Multiplication Principle cont’d Addition Principle

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13 Decision Algorithm A decision algorithm is a procedure in which we make a sequence of decisions. We can use decision algorithms to determine the number of possible items by pretending we are designing such an item (for example, an ice-cream cone) and listing the decisions or choices we should make at each stage of the process. Decision Algorithms: The Addition and Multiplication Principles

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14 Quick Example An iPod is available in two sizes. The larger size comes in two colors and the smaller size (the Mini) comes in four colors. A decision algorithm for “designing” an iPod is: Alternative 1: Select Large: Step 1 Choose a color: Two choices (so, there are two choices for Alternative 1.) Alternative 2: Select a Mini: Step 1 Choose a color: Four choices (so, there are four choices for Alternative 2.) Thus, there are 2 + 4 = 6 possible choices of iPods. Decision Algorithms: The Addition and Multiplication Principles

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