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7! = 5040 - Simpleprint

📅 March 14, 2026 👤 scraface

Mar 14, 2026

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📌 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.

📌 Let \( G \) be the graph with 5 vertices and 3 edges. Let \( d(v) \) be degree of \( v \).

📌 A path of length 2 is determined by a middle vertex and two neighbors.

📌 But we need the path to have *exactly* two edges in \( E \), and the three vertices involved fully described.

📌 So for each vertex \( v \), the number of paths of length 2 with center \( v \) is \( inom{\deg(v)}{2} \), and each such path \( u-v-w \) uses edges \( uv \) and \( vw \), both in \( E \). Then the trio \( \{u,v,w\} \) has exactly two close pairs (if no other edge), and since \( u \) and \( w \) are not necessarily connected, itâs exactly 2.

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