= Triangular number pair-counting identity
{title2=$\binom{N}{2}=T_{N-1}$}
For $N\geq1$ distinct objects, the number of <unordered pairs> of different objects is
$$
\binom N2=\frac{N(N-1)}2=T_{N-1}.
$$
Ordering the objects gives $N-1$ pairs with the first object, $N-2$ new pairs with the second, and so on down to one. Alternatively, $N(N-1)$ <ordered pairs> count every <unordered pair> twice. The <binomial coefficient> therefore counts the <edges> of a <complete graph> on $N$ <vertices>. This is also consistent with the <handshaking lemma>: each <vertex> has <degree of a vertex> $N-1$, and the degree sum counts every <edge> twice.
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