4-4 Counting 183 PERMUTATIONS RULE When n different items are available and r of them are selected without replacement, the number of different permutations (order counts) is given by nPr = n! 1n - r2! Permutations Rule (with Different Items): Trifecta Bet EXAMPLE 3 In a horse race, a trifecta bet is won by correctly selecting the horses that finish first and second and third, and you must select them in the correct order. The 144th running of the Kentucky Derby had a field of 20 horses. a. How many different trifecta bets are possible? b. If a bettor randomly selects three of those horses for a trifecta bet, what is the probability of winning by selecting Justify to win, Good Magic to finish second, and Audible to finish third, as they did? Do all of the different possible trifecta bets have the same chance of winning? (Ignore “dead heats” in which horses tie for a win.) YOUR TURN. Do Exercise 11 “Scheduling Routes.” SOLUTION a. There are n = 20 horses available, and we must select r = 3 of them without replacement. The number of different sequences of arrangements is found as shown: nPr = n! 1n - r2! = 20! 120 - 32! = 6840 b. There are 6840 different possible arrangements of 3 horses selected from the 20 that are available. If one of those arrangements is randomly selected, there is a probability of 1>6840 that the winning arrangement is selected. There are 6840 different possible trifecta bets, but not all of them have the same chance of winning, because some horses tend to be faster than others. (Because the “favorite” horses all finished in the top three, a winning $2 trifecta bet in this race won only $282.80.) 4. Permutations Rule (When Some Items Are Identical to Others) When n items are all selected without replacement, but some items are identical, the number of possible permutations (order matters) is found by using the following rule. PERMUTATIONS RULE (WHEN SOME ITEMS ARE IDENTICAL TO OTHERS) The number of different permutations (order counts) when n items are available and all n of them are selected without replacement, but some of the items are identical to others, is found as follows: n! n1!n2! . . . nk! wheren1 arealike, n2 are alike, . . . , andnk arealike. Bar Codes In 1974, the first bar code was scanned on a pack of Juicy Fruit gum that cost 67¢. Now, bar codes or “Universal Product Codes” are scanned about 10 billion times each day. When used for numbers, the bar code consists of black lines that represent a sequence of 12 digits, so the total number of different bar code sequences can be found by applying the fundamental counting rule. The number of different bar code sequences is 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 = 1012 = 1,000,000,000,000. The effectiveness of bar codes depends on the large number of different possible products that can be identified with unique numbers. When a bar code is scanned, the detected number is not price; it is a number that identifies the particular product. The scanner uses that identifying number to look up the price in a central computer. See the accompanying bar code representing the author’s name, so that letters are used instead of digits. There will be no price corresponding to this bar code, because this person is priceless—at least according to most members of his immediate family.
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