Symmetry Rule for Gaussian Binomial Coefficients

Theorem
Let $q \in \R_{\ne 1}, n \in \Z_{>0}, k \in \Z$.

Then:
 * $\dbinom n k_q = \dbinom n {n - k}_q$

where $\dbinom n k_q$ is a Gaussian binomial coefficient.

Proof
If $k < 0$ then $n - k > n$.

Similarly, if $k > n$, then $n - k < 0$.

In both cases:
 * $\dbinom n k_q = \dbinom n {n - k}_q = 0$

Let $0 \le k \le n$.

Consider the case $k \le \dfrac n 2$.

Then $k \le n - k$.

The case $k \ge \dfrac n 2$ can be done by observing:
 * $n - k \le \dfrac n 2$

and hence by the above:
 * $\dbinom n k_q = \dbinom n {n - \paren {n - k} }_q = \dbinom n {n - k}_q$