1977 AMC 12 第 20 题

Question

一条路径由若干水平和（或）竖直线段组成，每条线段连接下图中一对相邻字母。沿路径从起点走到终点时恰好拼出单词 CONTEST 的路径共有多少条？

For how many paths consisting of a sequence of horizontal and/or vertical line segments, with each segment connecting a pair of adjacent letters in the diagram below, is the word CONTEST spelled out as the path is traversed from beginning to end?

63

128

129

255

以上均不正确

none of these

答案：E

知识点：格路对称性基本计数

难度评级：2200

小提示：

将每条路径反向，从底行中央的

T

出发

Reverse each path, starting from the central

T

in the bottom row

大提示：

分别计算向左和向右的两类路径，再扣除它们共有的路径

Count the left-going and right-going families separately, then correct for their common path

解答：

把路径反向，从底部中央的

T

出发拼出 TSETNOC。对于水平移动均向左的一类路径，六步中的每一步都有向上或向左两种选择，因此共有

2^6=64

条路径。由对称性，水平移动向右的路径也有

64

条。完全竖直的中央路径同时属于两类，所以总数为

64+64-1=127\text{。}

这个数不在前四个选项中。

因此，正确答案是 E。

Reverse the paths and spell TSETNOC from the central bottom

T.

In the family whose horizontal moves go left, each of the six steps has two choices: up or left. This gives

2^6=64

paths. By symmetry,

64

paths have horizontal moves going right. The all-vertical central path belongs to both families, so the total is

64+64-1=127.

This number is not among the first four choices.

Therefore, the correct answer is E.

1977 AMC 12 第 20 题

答案：E

难度评级：2200

小提示：

大提示：

解答：

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