\begin{definition}[Definition:Gaussian Binomial Coefficient]
Let $q \in \R_{\ne 1}$, $r \in \R$, $m \in \Z_{\ge 0}$.
The '''Gaussian binomial coefficient''' is an extension of the more conventional binomial coefficient as follows:
{{begin-eqn}}
{{eqn | l = \binom r m_q
      | o = :=
      | r = \prod_{k \mathop = 0}^{m - 1} \dfrac {1 - q^{r - k} } {1 - q^{k + 1} }
      | c = 
}}
{{eqn | r = \dfrac {\paren {1 - q^r} \paren {1 - q^{r - 1} } \cdots \paren {1 - q^{r - m + 1} } } {\paren {1 - q^m} \paren {1 - q^{m - 1} } \cdots \paren {1 - q^1} }
      | c = 
}}
{{end-eqn}}
\end{definition}