  Equation of state with quantum statistics corrections is derived for a
multi-component gas of particles interacting through the repulsive and
attractive van der Waals (vdW) forces up to first few orders over a small
parameter $\delta \approx \hbar^3 n(mT)^{-3/2}[g(1- bn)]^{-1}$, where $n$ and
$T$ are the particle number density and temperature, $m$ and $g$ the particle
mass and degeneracy factor. The parameter $b$ corresponds to the vdW excluded
volume. For interacting system of Fermi nucleon and Bose $\alpha$ particles, a
small impurity of $\alpha$ particles to the nucleon system at leading first
order in both $\alpha $ particle and nucleon small parameters $\delta $ does
not change much the basic results for the symmetric nuclear matter. The
particle number fluctuations $\omega$ determined by the isothermal
in-compressibility $\mathcal{K}(n,T)$ can be obtained analytically at the same
first order quantum-statistics approximation for symmetric nucleon matter. Our
approximate analytical results appear to be in good agreement with the accurate
numerical calculations.
