1094 CHAPTER 11 Further Topics in Algebra To further illustrate the distinctions between permutations and combinations using tree diagrams, suppose we want to select 2 cans of soup from 4 cans. noodle 1N2, bean 1B2, mushroom 1M2, and tomato 1T2 As shown in Figure 15(a), there are 12 ways to select 2 cans from the 4 cans if order matters (if noodle first and bean second is considered different from bean, then noodle, for example). On the other hand, if order is unimportant, then there are 6 ways to choose 2 cans of soup from the 4 cans, as illustrated in Figure 15(b). Cream of Chicken & Mushroom Bean with Bacon Tomato Chicken & Noodles 1st Choice 2nd Choice Number of Ways N B M T B M T N M T N B T N B M 1 2 3 4 5 6 7 8 9 10 11 12 P(4, 2) = 12 (a) 1st Choice 2nd Choice Number of Ways N B M T B M T M T 1 2 3 4 5 T 6 C(4, 2) = 6 (b) Figure 15 CAUTION Not all counting problems lend themselves to either permutations or combinations. Whenever the fundamental principle of counting or a tree diagram can be used directly, as in the soup example, use it. CONCEPT PREVIEW Fill in the blank(s) to correctly complete each sentence. 1. From the two choices permutation and combination, a computer password is an example of a and a hand of cards is an example of a . 2. If there are 3 ways to choose a salad, 5 ways to choose an entrée, and 4 ways to choose a dessert, then there are ways to form a meal consisting of these three choices. 3. There are ways to form a three-digit number consisting of the digits 4, 5, and 9. 4. If there are 3 people to choose from, there are ways to choose a pair of them. 5. When a fair die is rolled and a fair coin is tossed,* there are possible outcomes. 6. A monogram consisting of three letters from the English alphabet can occur in different ways. 11.6 Exercises * A fair die has 6 faces with a different number of dots 1–6 on each face. A fair coin has 2 sides with heads on one side, tails on the other. In both cases, all outcomes have the same chance of occurring.
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