2025 AMC 8 第 25 题

Question

麦凯拉找出在一个

5 \times 5

菱形网格中画路径的所有可能方式。每条路径从网格底部开始，到顶部结束，每一步都向东北或西北移动一个单位。她计算每条路径与网格右边界之间区域的面积。下图给出了两个例子。所有可能路径确定的这些面积之和是多少？

Makayla finds all the possible ways to draw a path in a

5 \times 5

diamond-shaped grid. Each path starts at the bottom of the grid and ends at the top, always moving one unit northeast or northwest. She computes the area of the region between each path and the right side of the grid. Two examples are shown in the figures below. What is the sum of the areas determined by all possible paths?

2520

3150

3840

4730

5050

答案：B

知识点：格路对称性组合

难度评级：1930

小提示：

把每条路径与它的镜像路径配对

Pair every path with its mirror image

大提示：

一条路径和它的镜像所对应的两个区域会填满整个菱形

For a path and its mirror, the two regions fill the whole diamond

视频讲解：

Click to load, then click again to play

文字解答：

设所求和为

X

。

由对称性，如果改为求每条路径与左边界之间的面积和，结果也为

X

。

但若把这个和与原来的答案相加，就等于对所有路径，把左侧面积与右侧面积之和全部相加。

对每一条路径，这两个面积之和正好是

25

。

路径数等于重排

LLLLLRRRRR

的方式数，其中

L

表示向左，

R

表示向右。重排数是

10

选

5

，记为

\binom{10}{5}

。

所以

2X = 25 \times \binom{10}{5}

。两边除以

2

，得到

X

等于

\frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 \times 1} \times \frac{25}{2}\text{，}

也就是

3150

，选 B。

Let

X

be the answer.

By symmetry, if the question asked for the sum of areas between each path and the left side of the grid, then the answer would be exactly the same

X.

But if that answer is added to the original answer, that is exactly the same as the sum over all paths, of the sum of areas to the left and to the right.

For each path, that sum of areas is exactly

25.

The number of paths is equal to the number of ways to rearrange

LLLLLRRRRR,

where

L

stands for left and

R

stands for right, as the path walks up. The number of rearrangements is

10

choose

5,

denoted

\binom{10}{5}.

So,

2X = 25 \times \binom{10}{5}.

Dividing by

2,

we get that

X

equals

\frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 \times 1} \times \frac{25}{2},

which is

3150,

or choice B.