# 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:

 $\displaystyle \dfrac {1 - q^{r - k} } {1 - q^{k + 1} }$ $=$ $\displaystyle \dfrac {\left({1 - q^{r - k} }\right) / \left({1 - q}\right)} {\left({1 - q^{k + 1} }\right) / \left({1 - q}\right)}$ multiplying top and bottom by $1 - q$ $\displaystyle$ $=$ $\displaystyle \dfrac {\sum_{j \mathop = 0}^{r - k - 1} q^j} {\sum_{j \mathop = 0}^k q^j}$ Sum of Geometric Progression $\displaystyle \leadsto \ \$ $\displaystyle \lim_{q \mathop \to 1^-} \dfrac {1 - q^{r - k} } {1 - q^{k + 1} }$ $=$ $\displaystyle \lim_{q \mathop \to 1^-} \dfrac {\sum_{j \mathop = 0}^{r - k - 1} q^j} {\sum_{j \mathop = 0}^k q^j}$ $\displaystyle$ $=$ $\displaystyle \dfrac {\sum_{j \mathop = 0}^{r - k - 1} 1} {\sum_{j \mathop = 0}^k 1}$ $\displaystyle$ $=$ $\displaystyle \dfrac {r - k} {k + 1}$

Thus:

 $\displaystyle \lim_{q \mathop \to 1^-} \dbinom r k_q$ $=$ $\displaystyle \prod_{k \mathop = 0}^{m - 1} \dfrac {r - k} {k + 1}$ $\displaystyle$ $=$ $\displaystyle \prod_{k \mathop = 1}^m \dfrac {r - k - 1} k$ Translation of Index Variable of Product $\displaystyle$ $=$ $\displaystyle \dbinom r m$ Definition of Binomial Coefficient

$\blacksquare$