  We develop a first-principle approach to compute the counting statistics in
the ground-state of $N$ noninteracting spinless fermions in a general potential
in arbitrary dimensions $d$ (central for $d>1$). In a confining potential, the
Fermi gas is supported over a bounded domain. In $d=1$, for specific
potentials, this system is related to standard random matrix ensembles. We
study the quantum fluctuations of the number of fermions ${\cal N}_{\cal D}$ in
a domain $\cal{D}$ of macroscopic size in the bulk of the support. We show that
the variance of ${\cal N}_{\cal D}$ grows as $N^{(d-1)/d} (A_d \log N + B_d)$
for large $N$, and obtain the explicit dependence of $A_d, B_d$ on the
potential and on the size of ${\cal D}$ (for a spherical domain in $d>1$). This
generalizes the free-fermion results for microscopic domains, given in $d=1$ by
the Dyson-Mehta asymptotics from random matrix theory. This leads us to
conjecture similar asymptotics for the entanglement entropy of the subsystem
$\cal{D}$, in any dimension, supported by exact results for $d=1$.
