  The probability distribution of the number $s$ of distinct sites visited up
to time $t$ by a random walk on the fully-connected lattice with $N$ sites is
first obtained by solving the eigenvalue problem associated with the discrete
master equation. Then, using generating function techniques, we compute the
joint probability distribution of $s$ and $r$, where $r$ is the number of sites
visited only once up to time $t$. Mean values, variances and covariance are
deduced from the generating functions and their finite-size-scaling behaviour
is studied. Introducing properly centered and scaled variables $u$ and $v$ for
$r$ and $s$ and working in the scaling limit ($t\to\infty$, $N\to\infty$ with
$w=t/N$ fixed) the joint probability density of $u$ and $v$ is shown to be a
bivariate Gaussian density. It follows that the fluctuations of $r$ and $s$
around their mean values in a finite-size system are Gaussian in the scaling
limit. The same type of finite-size scaling is expected to hold on periodic
lattices above the critical dimension $d_{\rm c}=2$.
