Binomial Coefficient is instance of Gaussian Binomial Coefficient

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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

$\blacksquare$


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