Question

Suppose a graph has $26$ vertices and each pair of vertices is connected by exactly one edge. Find the number of edges in the graph. (Hint: If you use the Fundamental Counting Principle, be sure to count each edge only once.)

Short answer

There are total $45$ edges

Step-by-step solution

Step 1. Given information

A graph with 26 vertices is given

Step 2. Concept used

First find the total number of ways to draw an edge. Then divide it by two because each vertex is counted two times.

Step 3. Calculation

Total number of ways to draw an edge is-

$\begin{array}{l}{}^{10}P_2\\ =\frac{10!}{8!}\\ =90\end{array}$

Now divide it by two,

$\frac{90}{2}=45$