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\[ = 6n + 2 \] - 500apps

📅 March 13, 2026 👤 scraface

Mar 13, 2026

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📌 \frac{n}{n} \cdot \binom{n - k}{k} + (-1)^k \cdot \frac{n}{n - k} \cdot \binom{n - k - 1}{k - 1}

📌 But for small $n=8$, $k=3$, we use a constructive method.

📌 Fix one station as selected. By rotational symmetry, fix station 1. Then the two neighbors (8 and 2) cannot be selected. We choose 2 more from stations 3, 4, 5, 6, 7, with no two adjacent (since 4 and 5 are adjacent, etc.), and no adjacency to 1 or each other.

📌 So available: 3, 4, 5, 6, 7, excluding 2 and 8 → valid: 3–7.

📌 We need 2 non-adjacent from {3,4,5,6,7}.

📌 List all 2-element non-adjacent pairs in 5 consecutive positions:

📌 Total: 6 valid pairs with station 1 selected.

📌 But each selection is counted 3 times (once for each selected station), so total valid sets:

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