# 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:

$2! = 1 \times 2$

$\blacksquare$