172 CHAPTER 3 Probability Applications of Counting Principles The table summarizes the counting principles. Principle Description Formula Fundamental Counting Principle 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. m# n Permutations The number of permutations of n distinct objects n! The number of permutations of n distinct objects taken r at a time, where r … n nPr = n! 1n - r2! The number of distinguishable permutations of n objects where n1 are of one type, n2 are of another type, and so on, and n1 + n2 + n3 + g+ nk = n n! n1! # n2! gnk! Combinations The number of combinations of r objects selected from a group of n objects without regard to order, where r … n nCr = n! 1n - r2!r! Finding Probabilities A student advisory board consists of 17 members. Three members will be chosen to serve as the board’s chair, secretary, and webmaster. Each member is equally likely to serve in any of the positions. What is the probability of randomly selecting the three members who will be chosen for the board? SOLUTION Note that order is important because the positions (chair, secretary, and webmaster) are distinct objects. There is one favorable outcome and there are 17P3 = 17! 117 - 32! = 17! 14! = 17# 16# 15# 14! 14! = 17# 16# 15 = 4080 ways the three positions can be filled. So, the probability of correctly selecting the three members who hold each position is P1selecting the three members2 = 1 4080 ≈ 0.0002. You can check your answer using technology. For instance, using Excel’s PERMUT command, you can find the probability of selecting the three members, as shown at the left. TRY IT YOURSELF 6 A student advisory board consists of 20 members. Two members will be chosen to serve as the board’s chair and secretary. Each member is equally likely to serve in either of the positions. What is the probability of randomly selecting the two members who will be chosen for the board? Answer: Page A38 EXAMPLE 6 Study Tip To solve a problem using a counting principle, be sure you choose the appropriate counting principle. To help you do this, consider these questions. • Are there two or more separate events? Fundamental Counting Principle • Is the order of the objects important? Permutation • Are the chosen objects from a larger group of objects in which order is not important? Combination Note that some problems may require you to use more than one counting principle (see Example 8). EXCEL 4080 A 1 2 0.000245098 =PERMUT(17,3) =1/A2
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