SECTION 3.4 Additional Topics in Probability and Counting 169 Finding nPr Find the number of ways of forming four-digit codes in which no digit is repeated. SOLUTION To form a four-digit code with no repeating digits, you need to select 4 digits from a group of 10, so n = 10 and r = 4. nPr = 10P4 = 10! 110 - 42! = 10! 6! = 10# 9# 8# 7# 6! 6! = 5040 So, there are 5040 possible four-digit codes that do not have repeating digits. TRY IT YOURSELF 2 A psychologist shows a list of eight activities to a subject in an experiment. How many ways can the subject pick a first, second, and third activity? Answer: Page A38 Finding nPr Each year, 33 race cars start the Indianapolis 500. How many ways can the cars finish first, second, and third? SOLUTION You need to select three race cars from a group of 33, so n = 33 and r = 3. Because the order is important, the number of ways the cars can finish first, second, and third is nPr = 33P3 = 33! 133 - 32! = 33! 30! = 33# 32# 31# 30! 30! = 32,736. TRY IT YOURSELF 3 The board of directors of a company has 12 members. One member is the president, another is the vice president, another is the secretary, and another is the treasurer. How many ways can these positions be assigned? Answer: Page A38 In Example 3, note that the Fundamental Counting Principle can be used to obtain the same result. There are 33 choices for first place, 32 choices for second place, and 31 choices for third place. So, there are 33# 32# 31 = 32,736 ways the cars can finish first, second, and third. EXAMPLE 2 EXAMPLE 3 Tech Tip You can use technology such as Minitab, Excel, StatCrunch, or the TI-84 Plus to find the number of permutations of n objects taken r at a time. For instance, here is how to find nPr in Example 2 on a TI-84 Plus. Enter the total number of objects, n = 10. MATH Choose the PRB menu. 2: nPr Enter the number of objects taken, r = 4. ENTER TI-84 PLUS 10 nPr 4 5040 For help with operations with factorials and scientific notation, see Integrated Review at MyLab Statistics
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