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.

["Probability a Random Trio Has Exactly 2 Close Pairs in a Given Network of 5 Individuals", "In network analysis, understanding closeness among individuals helps uncover community structures and interaction patterns. In this article, we analyze a specific network of 5 individuals where exactly 3 pairs are "close" (linked), and we compute the probability that a randomly selected trio (an unordered group of 3 people) contains exactly 2 close pairs and 1 non-close pair.", "---", "### Scenario Setup", "We assume:", "- There are ( \binom{5}{2} = 10 ) total unordered pairs among the 5 individuals.\n- Among these, exactly 3 pairs are "close".\n- The remaining ( 10 - 3 = 7 ) pairs are non-close.", "We randomly select a trio of individuals — any 3-person group — and want the probability that this trio forms exactly 2 close pairs and 1 non-close pair.", "---", "### Step 1: Total Number of Tripolar Groups", "The number of unordered trios (groups of 3 people) from 5 individuals is:", "[\n\binom{5}{3} = 10\n]", "So there are 10 distinct groups to consider.", "---", "### Step 2: Count Favorable Trios — Those with Exactly 2 Close Pairs and 1 Non-Close Pair", "Let’s define what makes a trio qualify:\n- Among the 3 members, there are 3 total pairs.\n- Exactly 2 of these pairs are close, and 1 pair is non-close.\n- This implies the trio forms a structure where two of the three pairs are connected (close), and the third is not — forming a so-called “2-edge” configuration among 3 nodes.", "To count such trios:", "Idea: For each of the 3 close pairs, determine how many trios they can belong to and form valid 2-close-pair subgroups.", "But since the 3 close pairs are not fully specified (they could form different topologies), we assume a generic configuration consistent with the 3 close pairs — and note symmetry.", "---", "#### Key Insight: Count Trios with Exactly 2 Close Pairs", "Let ( G ) be a graph on 5 vertices with exactly 3 edges (the close pairs). There are many such graphs, but up to isomorphism, we analyze how many trios support exactly 2 edges.", "But instead of enumerating all 3-edge graphs on 5 nodes (there are ( \binom{5}{2} = 10 ) possible edge sets with 3 edges), we proceed via combinatorics conditioned on the structure and use symmetry.", "Alternatively, consider this:", "Each trio selects 3 vertices. Among them, there are 3 possible pairs. We want exactly 2 are close (i.e among the 3 edges in ( G )), and 1 is not.", "Suppose the 3-close-pair graph has structure such that some trios overlap nicely. But without knowing the actual configuration, we must determine if the probability depends only on global edge count and total trio configurations.", "However, since the problem states “a network with exactly 3 close pairs” — and no specification of topology — and since the answer must be computable, we assume the lowest symmetry overcount or average-case.", "But actually, the number of trios with exactly 2 close pairs depends on how the 3 edges are arranged.", "Let’s suppose instead a generic connected or nearly disconnected 3-edge graph.", "But here’s a better approach: fix all 3 edges among 5 nodes and count how many trios contain exactly two of them.", "We can compute this via inclusion and average-case reasoning.", "Let’s define:", "- Let ( T = \binom{5}{3} = 10 ): total trios\n- Let ( N ) = number of trios with exactly 2 close pairs and 1 non-close pair", "We compute ( N ) by analyzing potential configurations.", "But since not all 3-edge graphs on 5 nodes behave the same, consider a canonical example.", "---", "### Constructive Approach with a Sample Valid Network", "Suppose the 3 close pairs are: (A, B), (A, C), (B, C). That is, a triangle on A, B, C. The other 7 pairs: (A,D), (A,E), (A,F)? Wait — only 5 people.", "People: A, B, C, D, E.", "Suppose the 3 close pairs are: AB, AC, BC — forming a triangle on A,B,C.", "The other 7 pairs are all non-close: AD, AE, AF, BD, BE, CD, CE, DE — wait, total pairs: ( \binom{5}{2} = 10 ), so 3 close, 7 non-close.", "Now compute how many trios have exactly 2 close pairs.", "List all 10 trios:", "1. ABC: pairs AB, AC, BC → 3 close → not favorable\n2. ABD: AB (close), AD (non), BD (non) → 1 close → no\n3. ABE: AB (close), AE (non), BE (non) → 1 close → no\n4. ACD: AC (close), AD (non), CD (non) → 1 close → no\n5. ACE: AC (close), AE (non), CE (non) → 1 close → no\n6. ADE: AD, AE, DE → all non → 0 close → no\n7. BCD: BC, CD, BD → BC close, others non → exactly 1 close → no\n8. BCE: BC, BE, CE → all non → 0 close\n9. BDE: BD, BE, DE → all non → 0 close\n10. CDE: CD, CE, DE → all non → 0 close", "So none of the 10 trios have exactly 2 close pairs — all have 0 or more. In fact, only the ABC trio has 3, others have ≤1.", "So in this configuration, ( N = 0 ), probability is 0.", "But the problem implies it’s possible — so our assumption about the 3-edge graph must allow configurations with 2-close-pair trios.", "Try another configuration.", "Suppose the 3 close pairs are arranged as a pentalobe or a "star" with one isolated edge.", "Best idea: create a 3-edge graph where two trios share overlapping close pairs.", "Try a path: A–B–C, and A–D. So close pairs: AB, BC, AD.", "List all trios:", "1. ABC: AB, BC → 2 close; AC? no (non); AD? D not in → 2 close → yes\n2. ABD: AB, AD, BD? BD not close → AB, AD → 2 close → yes\n3. ACD: AC? no (A–C? no); AD, CD? CD not close → only AD → 1 close → no\n4. BCD: BC, CD? no → only BC → 1 close → no\n5. BCE: none → 0 → no\n6. BDE: none → no\n7. CDE: none → no\n8. ADE: AD, but AE, DE? no → only AD → 1 close → no\n9. ABC already done\nWait: list all 10:", "- ABC: AB, BC → 2 close; AC not close, AD? D not in → only AB, BC → 2 → yes\n- ABD: AB, AD → 2 close (since BC not in trio); BC not in, BD? not → only AB, AD → 2 → yes\n- ACD: AD, AC? no, CD? no → only AD → 1 → no\n- BCE: none → 0 → no\n- BDE: none → 0 → no\n- CDE: none → 0 → no\n- ACE: AC, AE, CE? none closed → 0 → no\n- BDA: same as ABD → AB, AD → 2 → yes (already counted)\nWait — order doesn’t matter.", "We must list unordered:", "Valid trios:", "1. ABC: closes AB, BC → 2 close → favorable\n2. ABD: closes AB, AD → 2 → favorable\n3. ACD: closes AD only → 1 → no\n4. BCE: none → 0 → no\n5. BDE: none → 0 → no\n6. CDE: none → 0 → no\n7. ABE: AB, AE? AE not close; BE? no; only AB → 1 → no\n8. ADE: AD, AE, DE? only AD close → 1 → no\n9. BCD: BC, CD? no; BD? no → only BC → 1 → no\n10. CDE: none → 0 → no", "Still only 2 trios with exactly 2 close pairs: ABC and ABD.", "Wait — is there a trio with 2 close and 1 non-close, different from these?", "Try a configuration where three edges form a triangle and one is connected.", "Let close pairs be: AB, BC, CA (triangle A-B-C), and AD.", "That’s same as before.", "Now try: close pairs: AB, CD, DE — disjoint.", "Then close edges: only these three, no overlap.", "Now check trios:", "- ABC: AB (close), but BC? no, AC? no → only 1 close → no\n- ABD: AB (close), BD? no, AD? not close (only AB, CD, DE) → AD? no → AB only → 1 close\n- ADE: AD? no (only DE), DE → 1\n- CDE: CD, DE, CE? no → 2 close? CD and DE → two close, but only two → so CDE has 2 close pairs → yes! CD and DE → both close → so CDE has 2 close pairs\nSimilarly:", "Trios:", "1. ABC: AB close, others? BC? no, AC? no → 1 close → no\n2. ABD: AB (close), BD? no, AD? no → 1 → no\n3. ACD: AC? no, CD? no → 0 → no\n4. ACE: AC? no, CE? no → 0 → no\n5. ADE: AD? no, DE → 1 → no\n6. BCD: BC? no, CD → 1 → no\n7. BCE: none → 0 → no\n8. BDE: BD? no, BE? no, DE → 1 → no\n9. CDE: CD, DE → 2 close → yes\n10. ADE? already\nWait: CDE has CD and DE → both close → so yes, 2 close → favorable", "Also: is there another?", "Try: ACD: AC? no, CD → only one close\nBCE: none\nBD? not close", "Only CDE has 2 close pairs.", "But we need exactly one trio with exactly 2 close pairs — but here only one trio has 2, but it has 2 closed pairs, others have ≤1 — but no trio has exactly 2 close pairs and 1 non? Wait, CDE has 2 close pairs and 0 non-close → not valid for “exactly 2 close, 1 non-close” — it has 2 close, but none non-close → not of required type.", "We need trios with exactly 2 close and 1 non-close.", "Still missing?", "Try a star plus an edge.", "Let close pairs: AB, AC, AD — star on A, connected to B,C,D. E isolated.", "Then close pairs: AB, AC, AD — total 3.", "Now check trios:", "1. ABC: AB, AC → 2 close; BC? no → so 2 close, 0 non → not valid (needs 1 non-close)\n2. ABD: AB, AD → 2 close, BD? no → same\n3. ACD: AC, AD → 2 close, CD? no → same\n4. BCE: BCE → only BC? none closed → 0 close\n5. BDE: no closed pair → 0\n6. CDE: none → 0\n7. ADE: AD, AE? no, DE? no → only AD → 1 close → no\n8. BCE → no\n9. BDE → no\n10. CDE → no\n11. ABC → same", "Only ABC, ABD, ACD each have 2 close pairs, but 0 non-close.", "No trio has exactly 1 non-close.", "Still not working.", "Wait — try a configuration with overlapping close pairs between trios.", "Suppose: close pairs: AB, BC, BD.", "So triangle B-C-D, plus A-B.", "So: AB, BC, BD all close.", "Now list trios:", "1. ABC: AB, BC → 2 close; AC? no → 2 close, 0 non → not valid\n2. ABD: AB, BD → 2 close, AD? no → 2 close → not valid (no non-close)\n3. ACD: AC? no, CD? no, AD? no → 0 close\n4. BCE: none → 0\n5. BDE: BD, DE? DE not close → only BD → 1\n6. CDE: none\n7. ABD → already\nWait — trio BDE: BD, but BE? no, DE? no → only BD → 1 close\nTrio ADE: AD, AE, DE? DE not (no), AD? no → 0\nTrio AEC: AC, AE, CE? 0\nTrio AED: same", "Only ABC and ABD have 2 close pairs — both have no non-close pair.", "So in all configurational attempts with exactly 3 close pairs, no trio has exactly 2 close pairs and 1 non-close pair?", "But that can’t be — unless the probability is zero.", "Wait — is it possible?", "Try this: close pairs: AB, AC, BD.", "So: A connected to B and C, B connected to D.", "So: AB, AC, BD closed.", "Now check trios:", "1. ABC: AB, AC → 2 close; BC? no, BC not close → no non-close → 2 close only\n2. ABD: AB, BD → 2 close; AD? no → 2 close\n3. ACD: AC, CD? CD? no → 1 close\n4. BCD: BC? no, BD → 1 close\n5. ADE: AD? no, AE? no, DE? no → 0\n6. BCE: none → 0\n7. BDE: BD → 1 close\n8. CDE: none\n9. ACE: AC → 1\n10. ADE: 0", "Now: Trios with exactly 2 close pairs: ABC, ABD.", "But both lack a third edge — so no non-close pair.", "So none satisfy exactly 2 close and 1 non-close.", "But we need such trios.", "Suppose we allow a trio to have 3 close pairs — but we want exactly 2.", "Can a trio have 2 close and 1 non-close?", "That requires the subgraph on 3 nodes to have exactly 2 edges among the 3 possible, and the third edge missing.", "So the 3-edge global graph must contain at least one trio that has exactly 2 of its 3 possible pairs as close, and the third not — i.e., the subgraph induced by the trio has exactly 2 edges.", "So we need a 3-edge graph on 5 nodes such that at least one trio has exactly 2 edges (i.e., 2 close pairs) and the third pair is non-close.", "This requires that the 3 edges do not form a triangle (which would give 3 close pairs), and are not disjoint.", "Example: place a triangle on A,B,C but remove one edge — no, we need exactly 3 close pairs.", "Wait — 3 close pairs define a graph on 5 vertices with 3 edges.", "Can such a graph contain a path of 3 edges? No, only 3 edges.", "Example: edges AB, BC, DE. Then close pairs: AB, BC, DE.", "Now check trios:", "- ABC: AB, BC → 2 close; AC? no → so two close, one non → exactly 2 close, 1 non-close → favorable!\n- ADE: DE → only one close (if AB,BC not in) → AD? no → only DE → 1 close\n- CDE: DE → only one\n- ABD: AB, BD? no, AD? no → only AB → 1\n- ACD: AC? no\n- BCE: none\n- BDE: BD? no, BE? no, DE → 1\n- AEC: no\n- So only ABC has exactly 2 close pairs and 1 non-close (AC, BC not both? BC is close, AC not) — in ABC: AB close, BC close, AC not → 2 close, 1 non → yes", "Are there others with 2 close pairs?", "Trio BDE: only DE → 1 close\nTrio CDE: only DE → 1", "Trio ACD: AC? no, CD? no → 0", "Trio ADE: only DE → 1", "Trio ABD: AB → 1", "Trio BCE: 0", "Trio BDE: only DE → 1", "Trio AEC? no edge? etc.", "But is there another trio with exactly 2 close pairs?", "Try BCD: BC? yes (close), CD? no, BD? no → only 1 close", "Only ABC has 2 close pairs and 1 non-close.", "So only one such trio.", "Total trios: 10", "So number of favorable: 1", "Thus, probability = ( \frac{1}{10} )", "But is this the only configuration?", "Suppose: close pairs: AB, AC, AD — star at A.", "Then: AB, AC, AD — 3 edges.", "Check trios with exactly 2 close pairs:", "- ABC: AB, AC → 2 close; BC? no → yes\n- ABD: AB, AD → 2 close → yes\n- ACD: AC, AD → 2 close → yes", "So three such trios.", "But now, do any of them have a non-close pair? No — all pairs are among AB, AC, AD — all close → each trio has 3 close pairs, not 2.", "So no trio has exactly 2 close pairs — only 3 or 0.", "So in this case, ( N = 0 )", "So probability depends on network structure.", "But the problem asks to compute the probability, implying it’s uniquely determined.", "Contradiction.", "Unless — the network is random, or we assume uniform distribution over such graphs?", "But the problem says: “We are given a network” — static.", "But since multiple topologies yield different counts, unless the probability is constant, the problem is ill-posed.", "But wait — perhaps the answer is the same for all 3-edge graphs on 5 vertices?", "From above:", "- In the path ABC with BD: only one trio ABC has 2 close pairs, but no non-close → 0 favorable\n- In the star: no trio has exactly 2 close pairs (only 3), and no non-close → 0\n- Try: close pairs: AB, CD, CE — a triangle B,C,D? No.", "Let: close pairs: AB, CD, CE", "So: A-B, C-D, C-E.", "Edges: AB, CD, CE — 3 edges.", "Now list trios:", "1. ABC: AB (close), AC? no, BC? no → 1 close\n2. ABD: AB (close), AD? no, BD? no → 1\n3. ACE: AC? no, CE (close), AE? no → AB? no → only CE → 1\n4. CDE: CD, CE, DE? no → CD, CE → 2 close; DE? no → so 2 close, 0 non → not favorable\n5. BCD: BC? no, CD → 1\n6. BCE: BC? no, CE → 1\n7. ADE: AD? no, AE? no, DE? no → 0\n8. BDE: BD? no, BE? no, DE? no → 0\n9. ACD: AC? no, CD → 1\n10. ABE? AB, AE, BE → only AB close → 1", "No trio has exactly 2 close and 1 non — closest is 1.", "Another try: close pairs: AB, BC, CD — a path of 4.", "Nodes: A-B-C-D, E isolated.", "Close: AB, BC, CD.", "Now trios:", "1. ABC: AB, BC → 2 close; AC? no →"]