11.8 Combinations 733 In Section 11.7, you learned that Pn r represents the number of permutations when r items are selected from n distinct items. Similarly , Cn r represents the number of combinations when r items are selected from n distinct items. b) The order in which two club members are selected to attend a meeting does not matter. For example, selecting Opal and then David is the same as selecting David and then Opal. Since the order of the selection is not important, this is a combination. 7 Now try Exercise 5 Did You Know? Lotteries and Probabilities Lotteries are a very popular form of gambling in the United States. Two of the largest lotteries, Powerball and Mega Millions, are available throughout most of the country. Many people dream of getting rich quickly by winning the lottery! The largest lottery jackpot in the world was a Powerball jackpot of $2.04 billion that was won by a single person from California in November 2022. While such a jackpot provides fuel for the imagination of winning big, just how likely are you to win the jackpot of one of these lotteries? Winning the Powerball jackpot requires matching 5 numbers chosen from 1 through 69 and then matching an extra number (the Powerball) from 1 through 26. The probability of winning the Powerball jackpot is 1 in 292,201,338. Winning the Mega Millions jackpot requires matching 5 numbers chosen from 1 through 70 and then matching the extra number (the Mega Ball) from 1 through 25. The probability of winning the Mega Millions jackpot is 1 in 302,575,350. You can verify both probabilities using combinations and the fundamental counting principle. For comparison, the probability of being dealt a royal flush in poker is 1 in 2,598,960. This means that you are about 112 times more likely to be dealt a royal flush than you are to win the Powerball lottery and about 116 times more likely to be dealt a royal flush than you are to win the Mega Millions lottery! Consider the set of elements a b c d e { , , , , }. The number of permutations of two letters from the set is represented as P , 5 2 and the number of combinations of two letters from the set is represented as C . 5 2 Twenty permutations of two letters and 10 combinations of two letters are possible from these five letters. Thus, P 20 5 2 = and C 10, 5 2 = as shown. Permutations Combinations ab ba ac ca ad da ae ea bc cb bd db be eb cd dc ce ec de ed , , , , , , , , , , , , , , , , , , , 20 ⎫ ⎬ ⎪ ⎭⎪ ab ac ad ae bc bd be cd ce de , , , , , , , , , 10 ⎫ ⎬ ⎪ ⎭⎪ When discussing both combination and permutation problems, we always assume that the experiment is performed without replacement. That is why duplicate letters such as aa or bb are not included in the preceding example. Note that from one combination of two letters, two permutations can be formed. For example, the combination ab gives the permutations ab and ba , or twice as many permutations as combinations. Thus, for this example we may write P C 2 ( ) 5 2 5 2 = ⋅ Since 2 2!, = we may write P C 2!( ) 5 2 5 2 = If we repeated this same process for comparing the number of permutations in Pn r with the number of combinations in C , n r we would determine that P r C !( ) n r n r = Dividing both sides of the equation by r! gives C P r! n r n r = Since P n n r ! ( )! , n r = − the combination formula may be expressed as C n n r r n n r r !/ ( )! ! ! ( )! ! n r = − = − Combination Formula The number of combinations possible when r objects are selected from n objects is determined by the combination formula C n n r r ! ( )! ! n r = − AlyssonM/Shuttersock
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