The tutor introduces the idea of the complete graph.

The complete graph on n vertices, K_n, means the graph in which each vertex is connected to each other one.

When two vertices are connected by an edge, they can be called adjacent. Therefore, the complete graph K_n has n vertices, each adjacent to all the others.

With K_n, the degree of each vertex (meaning the number of other vertices to which it is adjacent) is n-1. The number of edges, |E|, for K_n can be gotten two ways:

1. |E| = nC2. Reason: each edge joins a pair of vertices. Therefore, for K_n, the number of edges is the number of possible pairs of vertices, which is nC2.
2. For any graph, the sum of the degrees of the vertices is twice the number of edges: ∑deg(v) = 2|E|. Then, for K_n, ∑deg(v)=n(n-1)=2|E|, which means
   n(n-1)/2 = |E|.

Reassuringly, nC2 = n(n-1)/2.

HTH:)

Source:

Grimaldi, Ralph P. Discrete and Combinatorial Mathematics. Don Mills: Addison-
  Wesley, 1994.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.