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.

["Understanding Path Counts in Complete Graphs: How Many Paths of 3 Nodes Exist?", "When studying graph theory, one fundamental concept is that of a path—a sequence of nodes connected by edges without repeating vertices. A path of 3 distinct nodes forms a simple linear structure with exactly two endpoints and one middle node connecting them. Understanding how many such paths exist in a complete graph is essential for grasping basic combinatorial structures in networks.", "---", "### A 3-Node Path: Structure and Counting", "In graph theory terms, a path of three nodes is a sequence like ( A - B - C ), where each adjacent pair is connected by an edge, and no node repeats. Since the order of nodes matters only in the sequence (unless further symmetry is considered), we first count how many such ordered paths exist, then adjust for overcounting based on symmetry.", "#### Step 1: Choosing 3 Distinct Nodes\nIn a complete graph with 5 nodes, denoted ( K_5 ), every node is directly connected to every other node. To form a 3-node path:", "[\n\ ext{Number of ways to choose 3 nodes from 5} = \binom{5}{3} = 10\n]", "Each such selection gives us 3 distinct nodes.", "---", "### Inducing Paths and Avoiding Redundancy", "Since ( K_5 ) is complete, any 3 chosen distinct nodes form a triangle—a 3-cycle. But we are interested in paths, i.e., linear orderings with a middle node.", "For every triplet of distinct nodes ( {A, B, C} ), there are two paths of length 2 that use all three nodes:", "1. ( A - B - C )\n2. ( C - B - A )", "However, unless we fix a direction or orientation (which we rarely do in unoriented path counting), these two represent the same undirected path structure — just traversed in opposite directions.", "But in counting distinct path sequences (as often required in algorithms and network analysis), we usually treat ( A-B-C ) and ( C-B-A ) as distinct if direction matters. However, in most combinatorial definitions—especially in undirected graphs—such paths are considered unique when they visit the nodes in different orders.", "For consistent outcomes in applications, we count all permutations of 3 distinct nodes that form linear paths.", "---", "### Total Number of 3-Node Paths", "For each of the ( \binom{5}{3} = 10 ) node combinations:", "- The number of ways to arrange 3 nodes in a linear sequence (i.e., linear permutations) is ( 3! = 6 ).\n- However, only 2 of these 6 permutations form a valid path (with a middle node): ( A-B-C ) and ( C-B-A ). The others, like ( A-C-B ) or ( B-A-C ), are not monotonic—though still paths, they are symmetric reverses.", "If we define a distinct path by edge sequence (not just node labels), then each unordered triplet generates two directed paths: forward and backward.", "But in unweighted, undirected graphs, unless labeled direction matters, we typically count each linear ordering once.", "However, standard combinatorics counts number of node sequences that form paths: for 3 distinct nodes, each permutation that visits them in a line counts — but only two directions represent a linear simple path without cycles.", "But observe: each set of 3 nodes forms exactly two linear paths — one in each direction.", "Thus, total number of directed 3-node paths (ordered sequences with a middle node) among 5 nodes is:", "[\n\ ext{Number of triplets} \ imes \ ext{Number of directions per triplet} = \binom{5}{3} \ imes 2 = 10 \ imes 2 = 20\n]", "Alternatively, if we consider unordered node sets and ask: how many distinct paths of 3 nodes exist?, the answer is the number of linear arrangements, i.e., ordered sequences with a clear middle node.", "But best clarity:\n- For every combination of 3 distinct nodes, there are 2 distinct linear paths (forward and backward).\n- Therefore, total number of such paths in ( K_5 ) is:", "[\n\binom{5}{3} \ imes 2 = 20\n]", "---", "### But Wait: The Problem Focuses on Structure of the Path Itself", "The original prompt specifies:", "> “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: ( \binom{5}{3} = 10 ), each induces a path unless it's complete or linear.”", "Clarification:\n- “Induces a path” — all sets of 3 distinct nodes induce a path (since ( K_5 ) includes all edges), but only linear ones are simple paths.\n- However, every triplet forms a triangle — so all are cycles — but a path in graph theory is acyclic by definition.", "Thus, strictly:\n- The only simple paths of length 2 in ( K_5 ) with 3 nodes are the linear ones: ( A-B-C ), not cycles.\n- But in ( K_5 ), all edges exist — so every 3-node subset forms a triangle, but still supports the path ( A-B-C ), just not as a cycle.", "So yes — every 3-node subset induces exactly one undirected path of 3 nodes, but two when directionality matters.", "However, in standard interpretation, a path has two possible directions. The phrase “induces a path unless complete” likely refers to the fact that complete subgraphs on 3 nodes do contain all edges — but the path structure still exists uniquely as a linear sequence.", "Hence, interpreting “number of such paths” as distinct undirected paths of 3 nodes (i.e., unordered triples with a defined middle), the count is simply:", "[\n\binom{5}{3} = 10\n]", "But the second part says: “each induces a path unless it's complete or linear” — likely metaphorical.", "But re-reading: “Number of such paths in a complete graph of 5 nodes: number of ways to choose 3 distinct nodes: ( \binom{5}{3} = 10 ), each induces a path unless it's complete or linear.”", "This is ambiguous — but correct mathematical interpretation:", "- Total 3-node subsets: ( \binom{5}{3} = 10 )\n- Each subset defines exactly one simple path of 3 nodes (if considered as a sequence: middle, left, right)\n- Since ( K_5 ) includes all edges, every such triplet is a triangle, but the path exists structurally regardless.", "So the intended count — matching common problems — is 10 paths, one per 3-node combination, viewed as linear sequences.", "But wait: in ( K_5 ), a path of 3 distinct nodes corresponds to exactly one simple path graph ( P_3 ), and there are ( \binom{5}{3} = 10 ) such subgraphs — but each is isomorphic, yet distinct as subsets.", "In graph theory, when counting paths as induced subgraphs, the number of distinct path structures on 3 vertices in a complete graph is equal to the number of 3-node subsets, because each subset supports exactly one path of length 2.", "Thus, answer is:", "[\n\boxed{10}\n]", "Total number of simple paths of exactly 3 nodes (with one middle) in ( K_5 ): 10", "But more precisely:\n- Each ordered 3-node path (i.e., sequence ( A-B-C )): ( \binom{5}{3} \ imes 2 = 20 )\n- But number of unordered triples is 10, each supporting one undirected path structure\n- The phrase “each induces a path unless it's complete or linear” likely means:\n - All triplets are valid source graphs (not complete—though they are—but the path still exists)\n - “Unless linear” may be cautionary: if linear (i.e., no shortcuts), it’s still a path — no exclusion.", "But since all are trivalent and complete internally, the count stands:", "Final Answer: There are ( \binom{5}{3} = 10 ) such paths — one per 3-node combination — as unordered linear sequences.", "---", "### Summary", "- In a complete graph with ( n ) nodes, the number of 3-node paths (i.e., simple paths of length 2) equals the number of ways to choose 3 distinct nodes, since each induces a linear path when edges are present.\n- ( \binom{n}{3} ) gives the count. For ( n = 5 ): ( \binom{5}{3} = 10 ).\n- Each such triplet forms one path with two endpoints and a single middle node.\n- Direction may or may not be counted separately — in undirected graphs, each pair of directions represents the same unordered path structure, but often sequences are counted.\n- Hence, number of such paths is 10.", "---", "Keywords: path of 3 nodes, complete graph ( K_5 ), number of paths between nodes, graph theory, combinatorics of paths, ( \binom{5}{3} ), simple paths, linear sequences.", "Related Topics:\n- Tree paths\n- Subgraph traversal\n- Combinatorics of graphs\n- Undirected versus directed paths", "---", "For further reading: Explore how path counting changes in directed graphs or weighted networks."]