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Unformatted text preview: CS 173: Discrete Structures, Spring 2010 Homework 9 Solutions This homework contains 5 problems worth a total of 52 regular points. 1. Off to the races! [10 points] You decide to take in a day of races at the local horse racing track and are particularly interested in the action involving races number 1 and 2 for the day. In the first race there are 12 horses and in the second race there are 8. In both races the horses are given numbers, 1 through 12 in the first case and 1 through 8 in the second. At the conclusion of the race, the horses will be placed in the order they have finished. With photo finishes and other snazzy gadgets we assume that there is never a tie. In order to strategically use your money, you’ve decided to do a couple of calculations to see just how many different ways certain things can happen for some of the various bets. (a) How many different outcomes (i.e. orderings of all the horses) are there for each of the two races? How many possibilities for the sequence of two races? [Solution] 1st race: it’s a permutation of 12 items, or 12! outcomes. 2nd race: it’s a permutation of 8 items, or 8! outcomes. For each outcome of race 1, any outcome of race 2 will work, and vice versa. Thus in sequence, there are 12!8! outcomes considering both races. (b) A “trifecta” is a particularly well paying bet where you pick the first place, second place, and third place horses for a single race. Convinced that horse 4 will win first, horse 2 will win second and horse 10 will win third in the first race, you decide to buy a trifecta ticket to this effect. In how many different outcomes of the first race will you win money? [Solution] There are 12 horses in the first race, and we have fixed the positions of three of them. The number of matching outcomes is the number of ways we can order the remaining horses: 9! (c) If you pay twice as much money, you can “box” a trifecta and then you only have to pick the first three finishing horses without placing them in their specific order (this does not pay quite as much as a regular trifecta, unfortunately). If you are still convinced that the top 3 will be horses 4, 2, and 10, in how many different outcomes of the first race will you win money if you buy a trifecta box with those horses? [Solution] From the previous problem, we see that there are 9! matching outcomes whenever we set the positions of horse4, horse2, and horse10. There are 3! ways to order horse4/horse2/horse10 in the first three positions, so there are 3!9! matching out- comes. 1 (d) An “In the Money” bet on a given horse is a bet that the horse will come in either first, second, or third place. In the first race, you like horse number 4 and in the second race, horse number 8 looks particularly interesting....
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