Properties of Binomial Coefficients

This page gathers together some of the simpler and more common identities concerning binomial coefficients.

Particular Values

 * $\displaystyle \forall r \in \R: \binom r 0 = 1$


 * $\displaystyle \forall r \in \R: \binom r 1 = r$

These follow directly from the definition.

where $T_n$ is the $n$th triangular number.
 * $\displaystyle \forall n \in \Z, n \ge 0: \binom n 2 = T_n = \frac {r \left({r - 1}\right)} 2$

This is proved in Closed Form for Triangular Numbers: Proof using Binomial Coefficients.