Island Cities
There are five cities A, B, C, D, E on a certain island. Each city is connected to every other city by road. In how many ways can a person starting from city A come back to A after visiting some cities without visiting a city more than once and without taking the same road more than once? (The order in which he visits the cities also matters: e.g., the routes A - B - C - A and A - C - B - A are different.)
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1 solution
He can visit all 4 cities or 3 cities or 2 cities from B, C, D, E. If he visits all 4 cities = 4! = 24 If he visits 3 cities = 4C3.3! = 24 If he visits 2 cities = 4C2.2! = 12 Answer is therefore 60.