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Home » Temps total : \( rac300v + rac300v + 20 = 7 \). - RoadRUNNER Motorcycle Touring & Travel Magazine

Temps total : \( rac300v + rac300v + 20 = 7 \). - RoadRUNNER Motorcycle Touring & Travel Magazine

📅 May 1, 2026 👤 scraface

May 01, 2026

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📌 So star with 3 edges: only 3 close pairs, but no trio has exactly 2.

📌 Case 2: The three close pairs form a path: \( AB, BC, CD \). Edges: AB, BC, CD. Trio \( A,B,C \): AB, BC close â 2 close pairs. Trio \( B,C,D \): BC, CD close â 2 close pairs. Trio \( A,B,D \): AB, BD? BD not close, AD not close â only AB â 1 close â fails. Trio \( A,C,D \): AC not close â 0 â fails. So only the two middle trios (A,B,C) and (B,C,D) each have exactly 2 close pairs.

📌 So one two-edge path with distinct middle vertex.

📌 Number of such paths: a path of 3 nodes has 2 endpoints and one middle. Number of such paths in a complete graph of 5 nodes: number of ways to choose 3 distinct nodes: \( inom{5}{3} = 10 \), each induces a path unless it's complete or linear.

📌 But we fix the set of 3 close pairs. We need among all 3-edge sets with exactly two edges forming a path of length 2, how many such sets exist.

📌 Alternate approach: total number of trios with exactly two close pairs.

📌 Each such trio corresponds to a 2-edge path (i.e., two adjacent edges sharing a vertex, not forming a triangle).

📌 For each internal node of degree at least 2 in the close pair graph, we can count how many such paths go through it.

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