[img]2022m10x/ct_ve01204002j_0002_221018[/img]In the figure above, X and Y repr

游客2025-02-27 45

问题

In the figure above, X and Y represent locations in a district of a certain city where the streets form a rectangular grid. In traveling only north or east along the streets from X to Y, how many different paths are possible?

选项 A、720
B、512
C、336
D、256
E、56

答案 E

解析 Each possible path will consist of traveling a total of 3 grid segments north and 5 grid segments east. Thus, letting ’N’ represent traveling north by one grid segment and ’E’ represent traveling east by one grid segment, each path can be uniquely represented by an appropriate 8-character string of N’s and E’s. For example, as shown in the figure below, NEENEENE represents grid segments traveled in the order north, east, east, north, east, east, north, and east.

Therefore, the number of possible paths is equal to the number of appropriate 8-character strings of N’s and E’s, which is

.
since each appropriate string is determined when a specification is made for the 3 positions in the string at which the N’s are to be placed. Alternatively, the number of possible paths is equal to the number of permutations of 8 objects in which 3 are identical (the N’s) and the remaining 5 are identical (the E’s), and thus equal to

The correct answer is E.

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