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：

其他年份的第 20 题