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?

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 $126 \cdot 25 = 3150$.