= Solution
For a locally finite <discrete metric> space $(X,d)$ with basepoint $x_0$, the <growth function of a discrete metric space> is
$$
\beta_{X,x_0}(r)=|B(x_0,r)|
=\#\{x\in X:d(x,x_0)\leq r\}.
$$
In a <Cayley graph>, left multiplication by $gx_0^{-1}$ is a graph isometry carrying $x_0$ to any other vertex $g$. It therefore gives a bijection $B(x_0,r)\to B(g,r)$ for every $r$. Thus \b[the growth function of a fixed Cayley graph is exactly independent of the basepoint].
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