Question

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?

$2520$

$3150$

$3840$

$4730$

$5050$

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Written solution:

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.

2025 AMC 8 Problem 25

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Problem 25 in Other Years