Question

A, B, C and D are four towns any three of which are non-collinear. Then the number of ways to construct three roads each joining a pair of towns so that the roads do not form a triangle is

• A. 7
• B. 8
• C. 9
• D. 24

Hint:

In combinatorial geometry problems, list all possible configurations and eliminate the forbidden ones to count the valid arrangements.

Answer 1

The Correct Option is B

We have 4 towns, total possible roads: \( \binom{4}{2} = 6 \). We want to choose 3 roads such that they don’t form a triangle. A triangle forms if all three vertices are connected pairwise. Avoiding this means all roads must be connected to a central vertex (star shape). Number of such configurations: Choose central vertex in \(4\) ways, then choose any \( \binom{3}{3} = 1 \) set of roads from it to the others. This gives \(4\) configurations. Additionally, there are configurations where roads connect in a chain — counting gives total 8 ways.