4-4 Counting 185 SOLUTION a. There are n = 35 different numbers available, and we must select r = 5 of them without replacement (because the selected numbers must be different). Because order does not count, we need to find the number of different possible combinations. We get nCr = n! 1n - r2!r! = 35! 135 - 52!5! = 35! 30! # 5! = 324,632 b. If you select one 5-number combination, your probability of winning is 1>324,632. Typical lotteries rely on the fact that people rarely know the value of this probability and have no realistic sense for how small that probability is. This is why the lottery is often called a “tax on people who are bad at math.” YOUR TURN. Do Exercise 29 “Mega Millions.” Permutations or Combinations? Because choosing between permutations and combinations can often be tricky, we provide the following example that emphasizes the difference between them. Permutations and Combinations: Corporate Officers and Committees EXAMPLE 6 The Google company must appoint three corporate officers: chief executive officer (CEO), executive chairperson, and chief operating officer (COO). It must also appoint a Planning Committee with three different members. There are eight qualified candidates, and officers can also serve on the Planning Committee. a. How many different ways can the officers be appointed? b. How many different ways can the committee be appointed? YOUR TURN. Do Exercise 23 “Corporate Officers and Committees.” SOLUTION Note that in part (a), order is important because the officers have very different functions. However, in part (b), the order of selection is irrelevant because the committee members all serve the same function. a. Because order does count, we want the number of permutations of r = 3 people selected from the n = 8 available people. We get nPr = n! 1n - r2! = 8! 18 - 32! = 336 b. Because order does not count, we want the number of combinations of r = 3 people selected from the n = 8 available people. We get nCr = n! 1n - r2!r! = 8! 18 - 32!3! = 56 With order taken into account, there are 336 different ways that the officers can be appointed, but without order taken into account, there are 56 different possible committees. How to Choose Lottery Numbers Many books and suppliers of computer programs claim to be helpful in predicting winning lottery numbers. Some use the theory that particular numbers are “due” (and should be selected) because they haven’t been coming up often; others use the theory that some numbers are “cold” (and should be avoided) because they haven’t been coming up often; and still others use astrology, numerology, or dreams. Because selections of winning lottery number combinations are independent events, such theories are worthless. A valid approach is to choose numbers that are “rare” in the sense that they are not selected by other people, so that if you win, you will not need to share your jackpot with many others. The combination of 1, 2, 3, 4, 5, 6 is a poor choice because many people tend to select it. In a Florida lottery with $105 million in prizes, 52,000 tickets had 1, 2, 3, 4, 5, 6; if that combination had won, the top prize would have been only $1000. It’s wise to pick combinations not selected by many others. Avoid combinations that form a pattern on the entry card. Go Figure 43,252,003,274,489,856,000: Number of possible positions on a Rubik’s cube.
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