182 CHAPTER 4 Probability Permutations and Combinations: Does Order Count? When using different counting methods, it is essential to know whether different arrangements of the same items are counted only once or are counted separately. The terms permutations and combinations are standard in this context, and they are defined as follows: SOLUTION a. For the 5 different letters, the number of different arrangements is 5! = 5 # 4# 3# 2# 1 = 120. Note that this solution could have been done by applying the multiplication counting rule. The first letter can be any one of the 5 letters in “steam,” the second letter can be any one of the 4 remaining letters, and so on. The result is again 5# 4# 3# 2# 1 = 120. Use of the factorial rule has the advantage of including the factorial symbol, which is sure to impress. b. There is only one arrangement with the letters listed in alphabetical order (aemst), so the probability is 1>120. c. The letters of “steam” can be arranged to form these other words: teams, meats, mates, and tames, so there are five different words that can be formed. This is found by trial and error, and statistics is no help here. YOUR TURN. Do Exercise 1 “Notation.” DEFINITIONS Permutations of items are arrangements in which different sequences of the same items are counted separately. (The letter arrangements of abc, acb, bac, bca, cab, and cba are all counted separately as six different permutations.) Combinations of items are arrangements in which different sequences of the same items are counted as being the same. (The letter arrangements of abc, acb, bac, bca, cab, and cba are all considered to be the same combination.) Mnemonics for Permutations and Combinations ■ Remember “Permutations Position,” where the alliteration reminds us that with permutations, the positions of the items makes a difference. ■ Remember “Combinations Committee,” which reminds us that with members of a committee, rearrangements of the same members result in the same committee, so order does not count. 3. Permutations Rule (When All of the Items Are Different) The permutations rule is used when there are n different items available for selection, we must select r of them without replacement, and the sequence of the items matters. The result is the total number of arrangements (or permutations) that are possible. (Remember, rearrangements of the same items are counted as different permutations.) How Many Shuffles? After conducting extensive research, Harvard mathematician Persi Diaconis found that it takes seven shuffles of a deck of cards to get a complete mixture. The mixture is complete in the sense that all possible arrangements are equally likely. More than seven shuffles will not have a significant effect, and fewer than seven are not enough. Casino dealers rarely shuffle as often as seven times, so the decks are not completely mixed. Some expert card players have been able to take advantage of the incomplete mixtures that result from fewer than seven shuffles. A in r H m P f The Random Secretary One classical problem of probability goes like this: A secretary addresses 50 different letters and envelopes to 50 different people, but the letters are randomly mixed before being put into envelopes. What is the probability that at least one letter gets into the correct envelope? Although the probability might seem as though it should be small, it’s actually 0.632. Even with a million letters and a million envelopes, the probability is 0.632. The solution is beyond the scope of this text—way beyond. O c o g A a 5 l d l
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