Problem 19 · 2008 AMC 8
Medium
Counting & Probability
symmetric-counting
Eight points are spaced around at intervals of one unit around a 2 × 2 square, as shown. Two of the 8 points are chosen at random. What is the probability that the two points are one unit apart?
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Answer: B — 2/7.
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Hint 1 of 2
The 8 points sit evenly around the square's edge, so every point looks the same — whichever you pick first, it has the same number of 1-unit neighbors.
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Hint 2 of 2
Once you fix the first point, the question collapses to: of the other 7 points, how many are exactly 1 unit away?
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Approach: use symmetry — fix the first point, count favorable seconds
- By symmetry every point is interchangeable, so just pick any first point. Walking around the perimeter, exactly its two immediate neighbors are 1 unit away.
- That leaves 2 good choices out of the remaining 7 points, so the probability is 2/7.
- Why this transfers: when all starting choices are symmetric, condition on one of them — the messy "choose 2" count becomes a simple "favorable out of the rest."
Another way — count pairs directly:
- Total pairs of points: C(8,2) = 28. Adjacent (1-unit) pairs: the 8 points form a loop, so there are 8 neighboring pairs.
- Probability: 8/28 = 2/7.
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