914 CHAPTER 13 Counting and Probability Ordering such a meal requires three separate decisions: Choose an Appetizer Choose an Entrée Choose a Dessert 2 choices 4 choices 2 choices Look at the tree diagram in Figure 2. Note that for each choice of appetizer, there are 4 choices of entrées. And for each of these ⋅ = 2 4 8 choices, there are 2 choices for dessert. A total of ⋅ ⋅ = 2 4 2 16 different meals can be ordered. Figure 2 Cheesecake Ice cream Soup, chicken, ice cream Soup, chicken, cheesecake Cheesecake Ice cream Chicken Patty Liver Beef Chicken Patty Liver Beef Soup Salad Soup, patty, ice cream Soup, patty, cheesecake Cheesecake Ice cream Soup, liver, ice cream Soup, liver, cheesecake Cheesecake Ice cream Soup, beef, ice cream Soup, beef, cheesecake Cheesecake Ice cream Salad, chicken, ice cream Salad, chicken, cheesecake Cheesecake Ice cream Salad, patty, ice cream Salad, patty, cheesecake Cheesecake Ice cream Salad, liver, ice cream Salad, liver, cheesecake Cheesecake Ice cream Salad, beef, ice cream Salad, beef, cheesecake Dessert Entrée Appetizer Solution THEOREM Multiplication Principle of Counting If a task consists of a sequence of choices in which there are p selections for the first choice, q selections for the second choice, r selections for the third choice, and so on, the task of making these selections can be done in ⋅ ⋅ ⋅ p q r … different ways. Forming Codes How many two-symbol code words can be formed if the first symbol is an uppercase letter and the second symbol is a digit? Solution EXAMPLE 5 It sometimes helps to begin by listing some of the possibilities.The code consists of an uppercase letter followed by a digit, so some possibilities are A1, A2, B3, X0, and so on. The task consists of making two selections: The first selection requires choosing an uppercase letter (26 choices), and the second task requires choosing a digit (10 choices). By the Multiplication Principle, there are ⋅ = 26 10 260 different code words of the type described. Now Work PROBLEM 23 Example 4 demonstrates a general principle of counting.
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