Answer the questions below.(If necessary, consult a list of formulas.)(a) There are 13 European cities that Tony would eventually like to visit. On his next vacation, though, he only has time to visit 3 of the cities: one on Monday, one on Tuesday, and one on Wednesday. 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 13 European cities that Tony would eventually like to visit. On his next vacation, though, he only has time to visit 3 of the cities: one on Monday, one on Tuesday, and one on Wednesday. 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 Tony visits the cities matters. The formula for permutations is nPr = 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 = 13 (the total number of cities) and r = 3 (the number of cities Tony will visit).
So, the number of different schedules Tony can have is 13P3 = 13! / (13 - 3)! = 131211 = 1716.
(b) This is also a permutation problem because the order in which the committee members are chosen for the different roles matters.
In this case, n = 16 (the total number of committee members) and r = 4 (the number of roles to be filled).
So, the number of ways these 4 offices can be filled is 16P4 = 16! / (16 - 4)! = 161514*13 = 43680.
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