2025 AMC 8 Problem 25

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

Answer: B

Concepts:lattice paths symmetry combinations

Difficulty rating: 1930

Small hint:

Pair every path with its mirror image

Big hint:

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

Video solution:

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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.