Question

一只虫沿下图六边形网格中的线段从 A 走到 B。带箭头的线段只能沿箭头方向行走，并且虫子不会重复走同一条线段。共有多少条不同路径？

A bug travels from A to B along the segments in the hexagonal lattice pictured below. The segments marked with an arrow can be traveled only in the direction of the arrow, and the bug never travels the same segment more than once. How many different paths are there?

$2112$

$2304$

$2368$

$2384$

$2400$

答案：E

知识点：格路分类讨论乘法原理

难度评级：2460

解答：

按路径使用的反向箭头集合 $S$ 分类。若 $S=\varnothing$ ，路径由每一列选择一条前进边确定，共 $2\cdot2\cdot4\cdot4\cdot4\cdot2\cdot2=2^{10}$ 条。

若 $S$ 只含一个外侧反向箭头，每种外侧选择有 $2^8$ 条；若含两个外侧反向箭头但不含中间箭头，有 $2^6$ 条。

若 $S$ 只含中间反向箭头，有 $2^9$ 条；若含中间箭头和恰好一个外侧反向箭头，对每一种外侧箭头的选择都有 $2^7$ 条；若三个反向箭头都使用，有 $2^5$ 条。

总数为 $2^{10}$ $+2\cdot2^8$ $+2^6$ $+2^9$ $+2\cdot2^7$ $+2^5=2400$ 。

所以正确答案是 E。

Classify a path by the set $S$ of backward arrows it uses. If $S=\varnothing$ , the path is determined by choosing one forward arrow in each column, giving $2\cdot2\cdot4\cdot4\cdot4\cdot2\cdot2=2^{10}$ paths.

If $S$ uses only the left backward arrow, there are $2^8$ paths, and by symmetry the same for only the right backward arrow. If it uses both outer backward arrows but not the middle one, there are $2^6$ paths.

If $S$ uses only the middle backward arrow, there are $2^9$ paths. If it uses the middle arrow and exactly one outer backward arrow, there are $2^7$ paths for each choice of outer arrow. If it uses all three backward arrows, there are $2^5$ paths.

The total is $2^{10}$ $+2\cdot2^8$ $+2^6$ $+2^9$ $+2\cdot2^7$ $+2^5=2400$ .

Thus, E is the correct answer.

2012 AMC 10B 第 25 题

答案：E

难度评级：2460

解答：

其他年份的第 25 题