AMC 8 2017 Problem 15
AMC8 2017 年第 15 题
基本计数原理★★★☆☆
In the arrangement of letters and numerals below, by how many different paths can one spell AMC8? Beginning at the A in the middle, a path allows only moves from one letter to an adjacent (above, below, left, or right, but not diagonal) letter. One example of such a path is traced in the picture.
在下图所示的字母和数字的排布中，拼写出 AMC8 有多少种方法？要求从中间的 A 开始，移动的路径只允许从一个字母移到和这个字母相邻的字母（上，下，左，右，但不允许是对角线）。一种可能的路径标示在图中。
- A.8
- B.9
- C.12
- D.24
- E.36
正确答案 · Correct Answer
D
解析 · Solution
Solution 1
There are three different kinds of paths that are on this diagram. The first kind is when you directly count , , in a straight line. The second is when you count , turn left or right to get , then go up or down to count 8 and . The third is the one where you start with , move up or down to count , turn left or right to count , then move straight again to get 8. There are 8 paths for each kind of path, making for paths. Notice that the is adjacent to 4 s, each is adjacent to 3 s, and each is adjacent to 2 8's. So for each , there are 4 s, and for each , there are 3 s, and for each , there are 2 8's. Thus, the answer is
We can do this problem by computing how many ways there are to get to each letter (in order). There is 1 way to get to the in the center. We can only get to each of the other s by going there from the , so there is 1 way to get to each of the four s. For the s, we notice that four s are surrounded by one , and four s are surrounded by two s. Finally, each of the 8 s is surrounded by one with one way to get there and one with two ways to get there. Therefore, there are three paths to any of the 8 s. Since there are eight 8 s, the answer is
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