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n-1 = 19 - Crosslake

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 In a social network of 5 connected individuals, what is the probability that a randomly selected trio consists of exactly 2 individuals who share a close relationship and 1 who does not, given that exactly 3 pairs in the network are classified as "close"?

📌 We are given a network of 5 individuals with exactly 3 "close" pairs among the \( inom{5}{2} = 10 \) total pairs. We are to compute the probability that a randomly selected trio (unordered group of 3) includes exactly 2 close pairs and 1 non-close pair.

📌 Letâs proceed step by step.

📌 Step 1: Total number of trios

📌 inom{5}{3} = 10

📌 Step 2: Count number of trios with exactly 2 close pairs

📌 Therefore, for a fixed trio \( \{A, B, C\} \), suppose \( AB \) and \( BC \) are close, but \( AC \) is not. Then this trio contributes.

📌 So, to count such trios, we count the number of triples where two edges form a path of length 2 (i.e., a "V" shape), and the third edge is absent.

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