Definition of the Volume of a Solid Hamming Distance Sphere

Let $A$ be a $q$-ary alphabet and let $n\in \mathbb{N}$. Consider the space of words $A^n$ of length $n$ as a metric space with the Hamming distance. Then, given any word $w\in A^n$ and integer radius $ϵ\in \mathbb{N}$, the solid sphere $S_{ϵ}(w)$ has volume $\text{vol}(S_{ϵ}(w))=|S_{ϵ}(w)|$ given by the formula…

$$\begin{array}{rl}|S_{ϵ}(w)|& =\left(\genfrac{}{}{0ex}{}{n}{0}\right)+\left(\genfrac{}{}{0ex}{}{n}{1}\right)(q-1)+\left(\genfrac{}{}{0ex}{}{n}{2}\right)(q-1{)}^2+\dots +(\genfrac{}{}{0ex}{}{n}{ϵ})(q-1{)}^{ϵ}\\ & =\sum _{i=0}^{ϵ}\left(\genfrac{}{}{0ex}{}{n}{i}\right)(q-1{)}^i\end{array}$$