A,B,C and D are four towns any three of which are non- colinear. 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

Options

A. 7
B. 8
C. 9
D. 24

Answer

To solve the problem, we need to determine the number of ways to construct three roads between four towns (A, B, C, and D) such that the roads do not form a triangle.

### Step-by-Step Solution:

1. **Understanding the Problem**: We have four towns: A, B, C, and D. We need to connect these towns with three roads, ensuring that the roads do not form a triangle.

2. **Total Ways to Select Towns**: To construct three roads, we need to select any three towns from the four available towns. The number of ways to choose 3 towns from 4 is given by the combination formula:
\[ \binom{4}{3} = 4 \]
This means we can select any three towns in 4 different ways.

3. **Connecting the Selected Towns**: For each selection of three towns, we can connect them with roads. The number of ways to connect three towns (say A, B, and C) with three roads is the number of ways to arrange these roads. Since each town can connect to the other two, we can visualize this as a complete graph (K3) with three edges. The number of ways to arrange these roads is:
\[ 3! = 6 \]
However, since we are only interested in the connections and not the arrangement, we can simply say that there is 1 way to connect three towns without forming a triangle.

4. **Considering the Fourth Town**: The fourth town (D) can be connected to any of the three towns selected. If we connect D to any of the three towns, we will not form a triangle, as we are only adding one road to the existing three.

5. **Calculating Total Ways**: Now, we can calculate the total number of ways to construct the roads without forming a triangle. Since we have 4 ways to select the towns and for each selection, we have 1 way to connect them without forming a triangle:
\[ \text{Total ways} = 4 \times 1 = 4 \]

6. **Conclusion**: Therefore, the total number of ways to construct three roads such that they do not form a triangle is 4.

### Final Answer:
The number of ways to construct three roads, each joining a pair of towns so that the roads do not form a triangle is **4**.