Let $R$ be the skew polynomial ring $k_\mathfrak{q}[x_1,\ldots,x_m]$ where $x_ix_j=qx_jx_i$ with $q\in k^*$ and for all $i<j$.
The $q$ Multinomial Theorem states that $$(x_1+\ldots+x_m)^r=\Sigma_{r_1+\cdots+r_m=r;r_i\geq0}\binom{r}{r_1,\cdots,r_m}_qx_1^{r_1}\cdots x_m^{r_m}$$
Where $\binom{r}{r_1,\ldots,r_m}_q=\frac{[r]_q!}{[r_1]_q!\cdots[r_m]_q!}$, and $[l]_q!=\begin{cases}[l]_q[l-1]_q\cdots[1]_q\quad\mathrm{if}\;l\geq1\\1\quad\mathrm{if}\;l=0\end{cases}$, and $[l]_q=\frac{1-q^l}{1-q}$.
I could not find a proof of this fact online. Could you show me a proof or give me a reference?
Suppose $x_jx_i=q(x_ix_j)$ when $i<j$. The observation
$$ x_m(x_1+\cdots+x_{m-1})=q(x_1+\cdots+x_{m-1})x_m $$
means the $m=2$ case implies all of the $m>2$ cases by induction:
$$ \begin{array}{ll} (x_1+\cdots+x_m)^n & \displaystyle =\sum_{k=0}^n \left[\begin{matrix} n \\ k \end{matrix}\right](x_1+\cdots+x_{m-1})^{n-k}x_m^k \\ & = \displaystyle \sum_k \left[\begin{matrix} n \\ k \end{matrix}\right] \left(\sum \left[ \begin{matrix} n-k \\ k_1, \cdots,k_{m-1} \end{matrix}\right] x_1^{k_1}\cdots x_{m-1}^{k_{m-1}} \right)x_m^{k} \\ & = \displaystyle \sum \left[ \begin{matrix} n \\ k_1 , \cdots, k_m \end{matrix} \right]x_1^{k_1}\cdots x_m^{k_m}. \end{array} $$
There are numerous proofs of the $q$-binomial theorem online.