Number of edges of a complete graph

Statement

Let $n$ be a natural number. Then the complete graph $K_n$ has exactly $$\binom{n}{2} = \frac{n (n - 1)}{2}$$ edges.

Remarks

The edges of $K_n$ are exactly the $2$-element subsets of the vertex set. The number of k-element subsets gives the count. The handshaking lemma gives the same count: $n (n - 1) = 2 |E|$. Each simple graph on $n$ vertices has at most $\binom{n}{2}$ edges.

Explanations

L1 human

If $n$ people all shake hands with each other, there are $\frac{n(n-1)}{2}$ handshakes. Each person shakes $n - 1$ hands, and this counts each handshake twice.