132 CHAPTER 3 Probability The Fundamental Counting Principle In some cases, an event can occur in so many different ways that it is not practical to write out all the outcomes. When this occurs, you can rely on the Fundamental Counting Principle. The Fundamental Counting Principle can be used to find the number of ways two or more events can occur in sequence. If one event can occur in m ways and a second event can occur in n ways, then the number of ways the two events can occur in sequence is m# n. This rule can be extended to any number of events occurring in sequence. The Fundamental Counting Principle In words, the number of ways that events can occur in sequence is found by multiplying the number of ways one event can occur by the number of ways the other event(s) can occur. Using the Fundamental Counting Principle You are purchasing a new car. The possible manufacturers, car sizes, and colors are listed in the table. Manufacturer Car size Color Ford compact white (W) GM midsize red (R) Honda black (B) green (G) How many different ways can you select one manufacturer, one car size, and one color? Use a tree diagram to check your result. SOLUTION There are three choices of manufacturers, two choices of car sizes, and four choices of colors. Using the Fundamental Counting Principle, you can determine that the number of ways to select one manufacturer, one car size, and one color is 3# 2# 4 = 24 ways. Using a tree diagram, you can see why there are 24 options. Ford compact midsize W R B G W R B G GM compact midsize W R B G W R B G Honda compact midsize W R B G W R B G Tree Diagram for Car Selections TRY IT YOURSELF 3 You add another manufacturer, Toyota, and another color, tan, to the choices in Example 3. How many different ways can you select one manufacturer, one car size, and one color? Use a tree diagram to check your result. Answer: Page A38 EXAMPLE 3
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