Question

考虑在一个 $8 \times 8$ 方格中，从左下角沿网格线走到右上角、长度为 $16$ 的路径。求这样的路径中，恰好改变方向四次的路径数，如下图中的例子所示。

Consider the paths of length $16$ that follow the lines from the lower left corner to the upper right corner on an $8 \times 8$ grid. Find the number of such paths that change direction exactly four times, as in the examples shown below.

答案

答案：294

知识点：格路分拆与有序分拆

难度评级：2340

解答：

恰好改变方向四次的路径由五段最长的直线段组成，方向在向右与向上之间交替。若第一段向右，模式为 $R, U, R, U, R$ ：三段向右的正长度总和为 $8$ ，两段向上的正长度总和为 $8$ 。这样的组成数分别为 $\binom{7}{2} = 21$ 和 $\binom{7}{1} = 7$ ，给出 $21 \cdot 7 = 147$ 条路径。

从向上开始的路径对称地也有 $147$ 。条。总数为 $147 + 147 = 294$ 。

A path that changes direction exactly four times consists of five maximal straight runs, alternating between rightward and upward moves. If the first run is rightward, the pattern is $R, U, R, U, R:$ three rightward runs with positive lengths summing to $8,$ and two upward runs with positive lengths summing to $8.$ The counts of such compositions are $\binom{7}{2} = 21$ and $\binom{7}{1} = 7,$ giving $21 \cdot 7 = 147$ paths.

Paths starting upward are counted symmetrically, another $147.$ The total is $147 + 147 = 294.$

2024 AIME I 第 6 题

答案：294

难度评级：2340

解答：

其他年份的第 6 题