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Home » Solution: Two vectors are orthogonal if their dot product is zero. Compute: - Blask

Solution: Two vectors are orthogonal if their dot product is zero. Compute: - Blask

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 \binom{8}{3} = 56

📌 Now, count the number of ways to select 3 non-adjacent stations on a ring of 8. This is a circular arrangement problem where no two selected stations are next to each other.

📌 We use the known combinatorial result: the number of ways to choose $k$ non-adjacent positions from $n$ arranged in a circle is:

📌 \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}.

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