What is the maximum number of edges in a simple graph with n vertices. Justify your answer.

To find the maximum number of edges in a simple graph with n vertices, we need to consider the definition of a simple graph.

In a simple graph, there can be at most one edge between any two vertices, and there are no self-loops (edges connecting a vertex to itself).

To determine the maximum number of edges, we need to consider the number of possible pairs of vertices that can be connected by an edge.

In a simple graph with n vertices, each vertex can be connected to (n-1) other vertices, as it cannot be connected to itself.

Therefore, the maximum number of edges in a simple graph with n vertices can be calculated using the formula:

Maximum number of edges = (n * (n-1)) / 2

This formula accounts for the fact that each edge is counted twice (once for each vertex it connects) and divides by 2 to avoid double counting.

By simplifying the formula, we get:

Maximum number of edges = (n^2 - n) / 2

So, the maximum number of edges in a simple graph with n vertices is (n^2 - n) / 2.