Definition:Binomial Coefficient/Integers/Definition 3

Definition
Let $n \in \Z_{\ge 0}$ and $k \in \Z$.

Then the binomial coefficient $\dbinom n k$ is defined as the coefficient of the term $a^k b^{n - k}$ in the expansion of $\paren {a + b}^n$.

Also see

 * Equivalence of Definitions of Binomial Coefficient