(If necessary, consult a list of formulas.)(a) There are 12 European cities that Kira would eventually like to visit. On her next vacation, though, she only has time to visit 3 of the cities: one on Monday, one on Tuesday, and one on Wednesday. She is now trying to make a schedule of which city she'll visit on which day. How many different schedules are possible? (Assume that she will not visit a city more than once.)(b) 72 athletes are running a race. A gold medal is to be given to the winner, a silver medal is to be given to the second-place finisher, and a bronze medal is to be given to the third-place finisher. Assume that there are no ties. In how many possible ways can the 3 medals be distributed?
Question
(If necessary, consult a list of formulas.)(a) There are 12 European cities that Kira would eventually like to visit. On her next vacation, though, she only has time to visit 3 of the cities: one on Monday, one on Tuesday, and one on Wednesday. She is now trying to make a schedule of which city she'll visit on which day. How many different schedules are possible? (Assume that she will not visit a city more than once.)(b) 72 athletes are running a race. A gold medal is to be given to the winner, a silver medal is to be given to the second-place finisher, and a bronze medal is to be given to the third-place finisher. Assume that there are no ties. In how many possible ways can the 3 medals be distributed?
Solution
(a) This is a permutation problem because the order in which Kira visits the cities matters. The formula for permutations is P(n, r) = n! / (n-r)!, where n is the total number of options, r is the number of options chosen, and "!" denotes factorial.
In this case, n = 12 (the total number of cities) and r = 3 (the number of cities she will visit).
So, P(12, 3) = 12! / (12-3)! = 121110 = 1320.
So, there are 1320 different schedules Kira could follow.
(b) This is also a permutation problem because the order in which the athletes finish matters.
In this case, n = 72 (the total number of athletes) and r = 3 (the number of medals).
So, P(72, 3) = 72! / (72-3)! = 727170 = 359,640.
So, there are 359,640 different ways the 3 medals could be distributed.
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