Binomial Coefficient with Two/Corollary

Theorem

 * $\forall n \in \N: \dbinom n 2 = T_n = \dfrac {n \left({n - 1}\right)} 2$

where $T_n$ is the $n$th triangular number.

Proof
From the definition of binomial coefficient:
 * $\dbinom n 2 = \dfrac {n!} {2! \ \left({n - 2}\right)!}$

The result follows directly from the definition of the factorial:
 * $2! = 1 \times 2$

Also see

 * Closed Form for Triangular Numbers: Proof using Binomial Coefficients