Answer the questions below.(If necessary, consult a list of formulas.)(a) There are 14 European cities that Bob would eventually like to visit. On his next vacation, though, he only has time to visit 4 of the cities: one on Monday, one on Tuesday, one on Wednesday, and one on Thursday. He is now trying to make a schedule of which city he'll visit on which day. How many different schedules are possible? (Assume that he will not visit a city more than once.)(b) A certain committee consists of 16 people. From the committee, a president, a vice-president, a secretary, and a treasurer are to be chosen. In how many ways can these 4 offices be filled? Assume that a committee member can hold at most one of these offices.
Question
Answer the questions below.(If necessary, consult a list of formulas.)(a) There are 14 European cities that Bob would eventually like to visit. On his next vacation, though, he only has time to visit 4 of the cities: one on Monday, one on Tuesday, one on Wednesday, and one on Thursday. He is now trying to make a schedule of which city he'll visit on which day. How many different schedules are possible? (Assume that he will not visit a city more than once.)(b) A certain committee consists of 16 people. From the committee, a president, a vice-president, a secretary, and a treasurer are to be chosen. In how many ways can these 4 offices be filled? Assume that a committee member can hold at most one of these offices.
Solution
(a) This is a permutation problem because the order in which Bob 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 = 14 (the total number of cities) and r = 4 (the number of cities Bob will visit). So, we have:
P(14, 4) = 14! / (14-4)! = 14! / 10! = 141312*11 = 24,024
So, there are 24,024 different schedules possible for Bob's trip.
(b) This is also a permutation problem because the order in which the committee members are chosen for the different offices matters. In this case, n = 16 (the total number of committee members) and r = 4 (the number of offices to be filled). So, we have:
P(16, 4) = 16! / (16-4)! = 16! / 12! = 161514*13 = 43,680
So, there are 43,680 different ways to fill the 4 offices.
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