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Thinking Mathematically The Fundamental Counting Principle The Fundamental Counting Principle If you can choose one item from a group of M items and a second item from a group of N items, then the total number of two-item choices is M N. Example Applying the Fundamental Counting Principle The Greasy Spoon Restaurant offers 6 appetizers and 14 main courses. Using the fundamental counting principle, there are 14 6 = 84 different ways a person can order a two-course meal. Example Applying the Fundamental Counting Principle This is the semester that you decide to take your required psychology and social science courses. Because you decide to register early, there are 15 sections of psychology from which you can choose. Furthermore, there are 9 sections of social science that are available at times that do not conflict with those for psychology. In how many ways can you create two-course schedules that satisfy the psychology-social science requirement? Solution The number of ways that you can satisfy the requirement is found by multiplying the number of choices for each course. You can choose your psychology course from 15 sections and your social science course from 9 sections. For both courses you have: 15 9, or 135 choices. The Fundamental Counting Principle The number of ways a series of successive things can occur is found by multiplying the number of ways in which each thing can occur. Example Options in Planning a Course Schedule Next semester you are planning to take three courses - math, English, and humanities. Based on time blocks and highly recommended professors, there are 8 sections of math, 5 of English, and 4 of humanities that you find suitable. Assuming no scheduling conflicts, there are: 8 5 4 = 160 different three course schedules. Example Car manufacturers are now experimenting with lightweight three-wheeled cars, designed for a driver and one passenger, and considered ideal for city driving. Suppose you could order such a car with a choice of 9 possible colors, with or without air-conditioning, with or without a removable roof, and with or without an onboard computer. In how many ways can this car be ordered in terms of options? Solution This situation involves making choices with four groups of items. color - air-conditioning - removable roof - computer 9 2 2 2 = 72 Thus the car can be ordered in 72 different ways. Example A Multiple Choice Test You are taking a multiple-choice test that has ten questions. Each of the questions has four choices, with one correct choice per question. If you select one of these options per question and leave nothing blank, in how many ways can you answer the questions? Solution We use the Fundamental Counting Principle to determine the number of ways you can answer the test. Multiply the number of choices, 4, for each of the ten questions 4444444444 =1,048,576 Example Telephone Numbers in the United States Telephone numbers in the United States begin with three-digit area codes followed by seven-digit local telephone numbers. Area codes and local telephone numbers cannot begin with 0 or 1. How many different telephone numbers are possible? Solution We use the Fundamental Counting Principle to determine the number of different telephone numbers that are possible. 8 10 10 8 10 10 10 10 10 10 =6,400,000,000 Thinking Mathematically The Fundamental Counting Principle