Selecting three roads among 4 towns without forming a triangle: Four towns A, B, C, D (no three collinear). Choose 3 undirected roads (each joins a pair of towns) so that the chosen roads do not form a triangle. How many such selections are possible?

Introduction / Context:With 4 towns, the complete undirected graph K4 has 6 edges (roads). We must choose 3 edges that do not form a 3-cycle (triangle). This is a graph-subset counting exercise with exclusion of triangular 3-edge sets.

Given Data / Assumptions:

• Vertices: 4 towns; edges: all unordered pairs (6 total).
• Select exactly 3 edges.
• Disallow any 3-edge selection that is exactly a triangle on some triple of towns.

Concept / Approach:

• Total 3-edge subsets = C(6,3) = 20.
• Triangles in K4: each choice of 3 towns determines exactly 1 triangle; there are C(4,3) = 4 such triangles.
• Valid sets = total − triangles.

Step-by-Step Solution:

Verification / Alternative check:Enumerating non-triangular 3-edge graphs on 4 vertices matches 16: these are exactly the 3-edge forests or graphs containing at least one vertex of degree 3 without closing a cycle of length 3.

Why Other Options Are Wrong:

• 12, 14, 18 are off by miscounting triangles or edges.
• “None of these” is false because 16 is correct.

Common Pitfalls:

• Forgetting there are 4 distinct triangles in K4 (one per 3-vertex subset).

Final Answer:16