A local family is planning a summer vacation. After much deliberation, they have

Question

A local family is planning a summer vacation. After much deliberation, they have decided that the following tourists destinations are the ones they will choose from: Disney World, Mount Rushmore, Rocky Mountain National Park, the Space Needle, the Mall of America, the Statue of Liberty, Niagara Falls. For their vacation, they can choose any or all of the destinations (they have a lot of money and time on their hands), or they can choose to stay home to get some work done around the house. (a) How many vacations are possible? (b) How many vacations are possible if the family actually leaves Illinois? (c) How many vacations are possible if the family does not go to every destination? (d) How many vacations are possible if the family goes to at least two destinations? (e) Suppose now the family decides they can take five vacations during the year. If they can choose to visit a location more than once, in how many ways can the family take the five vacations? (f) In part (e), suppose the family decides they definitely want to go to Disney World and Rocky Mountain National Park each at least once. Now in how many ways can the family take five vacations?

Explanation / Answer

(a) They have none to seven number of options for tourist destinations.

Number of vacations possible = 27 -1 = 128 -1 = 127

(b) Number of vacations possible if the family leaves illinois = 26 -1 = 63

(c) Vacations possible when they dont go to every destination will be when we left each destination for the comination.

Number of vacations possible = 128 -1 - 1 = 126

() Number of vacation possible when they go to atleast 2 destinations = 128 - 7C0 - 7C1 = 120

(e) Number of ways to chose 5 locations during the year out of 7 = 7C5 = 21

(f) Here they have chosen atleast 2 places to visit, now out of 5 remaining places, they have to chose 3.

so number of such places ar e= 5C3 = 10

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