Problem 25 · 2025 AMC 8
Stretch
Counting & Probability
careful-countingsymmetry
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Answer: B — 3150.
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Hint 1 of 2
Summing the right-area of all 252 paths one at a time is hopeless. The escape: mirror each path left↔right. The full set of paths is its own mirror image, so the total of all left areas equals the total of all right areas — the answer you want.
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Hint 2 of 2
Add the two totals: for a single path, (left area) + (right area) is just the whole diamond, a constant. So 2×(answer) = (that constant) × (number of paths). The path count is the ways to interleave 5 NE and 5 NW moves.
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Approach: double it via the left↔right mirror, so each path contributes a constant
- Let X be the sum of right-side areas. Flipping every path left↔right sends the set of all paths to itself, so the sum of all left-side areas is also X. Add them: 2X = sum over all paths of (left + right).
- But for any single path, left area + right area is the entire 5×5 diamond = 25 — a constant, independent of the path. The hard per-path detail vanishes.
- Number of paths = interleavings of 5 NE moves and 5 NW moves = 10!5! · 5! = 252.
- So 2X = 25 × 252 = 6300, giving X = 3150.
- Why this transfers: to sum a quantity you can't compute term-by-term, add it to its mirror image — if the pair sums to a constant, you've turned an impossible sum into (constant) × (count) ÷ 2. This pairing/symmetry trick is the same idea behind summing 1+2+…+n by pairing ends.
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