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Unformatted text preview: Discrete Mathematics Math 6A Homework 5 Solution 5.16 There are 16 cards that qualify as being an ace or a heart, so The prob. that a card selected from a deck is an ace or a heart = 16/52 5.115 We have to choose five cards. That is, two pairs and another card that is not the same number with two pairs. There are four kinds of card and each kind has 13 cards. So the total number of cards is 52(=13*4) For example, we can choose 1 heart + 1 diamond + 2 heart + 2 diamond + another number that is not the same with 1 or 2 among four kinds. There are C(13,2)*C(4,2)*C(4,2)*(528) = 78*6*6*44=123,552 different hands. Since each hand is equally likely, and there are C(52,5) = 2,598,960 different hands. Therefore, the probability of holding two pairs is 123,552/ 2,598,960 5.118 A "straight flush" means choosing a consecutive five cards of the same kind among the four kinds. For example, we can choose 12345, 23456,..., 10,J,Q,K,A that is 10 ways for choosing the straight flush. So there are 10*4 straight flushes. Therefore, the prob. to choose the straight flush is 40/C(52,5) = 1/64974 Note: 10,J,Q,K,A is actually "Royal Straight Flush", but we count this as a straight flush for this problem. Note: From the J,Q,K,A,1 or Q,K,A,1,2 or etc are NOT the straight flush ! 5.127 First, we need to select exactly one of the correct six integers among the n numbers and the order does not matter. There are C(n,6) possible choices to pick the correct six integers and we need to select one of C(n,6). Second, there are n6 incorrect integers and we need to select rest of five incorrect integers among the C(n6,5). Third, there are six ways to choose exactly one correct integers among the six correct integers. So the prob. of selecting exactly one of the correct six integers is 6{C(n6,5)/C(n,6)}...
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 Winter '08
 JARVIS
 Math

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