Binomial Coefficient with Two/Corollary

Theorem

 * $\forall n \in \N: \dbinom n 2 = T_{n - 1} = \dfrac {n \paren {n - 1} } 2$

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

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

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