Binomial Coefficient is instance of Gaussian Binomial Coefficient

Theorem
Let $\dbinom r m_q$ denote the Gaussian binomial coefficient:

Then:
 * $\displaystyle \lim_{q \mathop \to 1^-} \dbinom r m_q = \dbinom r m$

where $\dbinom r m$ denotes the conventional binomial coefficient.

Proof
We have by definition of Gaussian binomial coefficient:
 * $\displaystyle \dbinom r m_q = \prod_{k \mathop = 0}^{m - 1} \dfrac {1 - q^{r - k} } {1 - q^{k + 1} }$

Consider a typical factor of this product:

Thus: