Makayla 在一个 $5 \times 5$ 的菱形网格中找到了所有可能的绘制路径的方式。每条路径从网格底部开始，到网格的顶部结束，总是朝东北或西北方向移动一个单位。她计算了每条路径与网格右侧边缘之间的区域面积。下图展示了两个示例。请问所有可能路径与网格右侧边缘之间的区域面积之和是多少？

A. 2520

B. 3150

C. 3840

D. 4730

E. 5050

1. 2025 AMC 8 官方解析

Answer (B): For each path, there is an area enclosed between the path and the left side of the grid, which is complementary to the area to the right of the path. Reflecting each path across a vertical line through the center of the grid swaps the roles of the areas to the left and right of the path.

This means the paths can be split into pairs so that the area enclosed by each pair of paths is 25. Because 25 is odd, no path will be paired with itself. Each path consists of 10 total steps, 5 of which are to the northeast and 5 of which are to the northwest. A path is determined by choosing the 5 positions of the northeast steps, so the total number of paths is ${}_{10}C_5 = 252$. Thus there are $\frac{252}{2} = 126$ pairs, each of which contributes 25 units of area. The sum of the areas determined by all the paths is $126 \cdot 25 = 3150$.