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9.2 - 9.2 The Multiplication Principle Permutations and...

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9.2 The Multiplication Principle, Permutations, and Combinations Counting plays a major role in probability. In this section we shall look at special types of counting problems and develop general formulas for solving them. We begin with an example that will demonstrate a general counting principle. 1. The fixed-price dinner at Bubba’s Barbeque provides the following choices: Appetizer: Soup or Salad Entrée: Chicken, Beef Brisket, Ribs, or Sausage Dessert: Apple Pie or Peach Cobbler How many different meals can be served up at Bubba’s? Choose an appetizer Choose an Entrée Choose a Dessert 2 choices 4 choices 2 choices Look at the tree diagram. We see that, for each choice of appetizer there are 4 choices of entrees. 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.

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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, then the task of making these selections can be done in ݌ · ݍ · ݎ · … different ways. 1. Joe has 5 shirts, 6 trousers, 3 ties, and 4 sport coats. How many different arrangements can he wear? Solution. By the counting principle he has 5 · 6 · 3 · 2 ൌ 180 different arrangements. 2. How many different 4-letter radio station call letters can be made a) if the first letter must be a K or W and no letter may be repeated? b) if repeats are allowed (but the first letter is a K or W) c) How many of the 4-letter call letters (starting with K or W) with no repeats end in R a) By the counting principle: 1 · 25 · 24 · 23 ൅ 1 · 25 · 24 · 23 ൌ 27,600 b) 1 · 26 · 26 · 26 ൅ 1 · 26 · 26 · 25 ൌ 35,152 c) 1 · 24 · 23 · 1 ൅ 1 · 24 · 23 · 1 ൌ 1104 3. The United States Postal Service currently uses 5-digit zip codes in most areas. How many zip codes are possible if there are no restrictions on the digits used? How many would be
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