A manufacturing company in city A wishes to truck its product to 4 different cities, B, C, D, and E. If the cities are all interconnected by roads, how many different route plans can be constructed so that a single truck, starting from A, will visit each city exactly once, then return home?
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3! = 3*2*1 = 12
Starting at A, he can go to B or C or D (three choices). From there, the driver has two choices. Then will have only one route left.
That sounds like an easy version of the standard traveling sales person problem.
To find the number of different route plans, we can use a concept from graph theory called Hamiltonian cycles. A Hamiltonian cycle is a path in a graph that visits every vertex exactly once and returns to the starting vertex. In this case, the cities are represented as vertices, and the roads connecting them are represented as edges.
We can approach this problem using a factorial calculation. Since the truck starts and ends in city A, there are only three remaining cities to visit: B, C, and D. We can calculate the number of arrangements of these three cities using the factorial function.
The factorial function, denoted by the exclamation mark (!), calculates the product of all positive integers from 1 up to a given number. For example, 3! (read as "3 factorial") is equal to 3 * 2 * 1 = 6.
So, the number of different arrangements of the three cities (B, C, D) can be calculated as 3!. Once we have the arrangement of these three cities, we fix the starting and ending city as city A. Therefore, for each arrangement of the three cities, there is one unique route plan.
Now, let's calculate the factorial:
3! = 3 * 2 * 1 = 6
So, there are 6 different route plans for the truck to visit each city once and return home.
To summarize:
1. Calculate the factorial of the number of remaining cities (in this case, 3!).
2. Use the result from step 1 as the number of different route plans.