  We consider the binomial random set model $[n]_p$ where each element in
$\{1,\dots,n\}$ is chosen independently with probability $p:=p(n)$. We show
that for essentially all regimes of $p$ and very general conditions for a
matrix $A$ and a column vector $\mathbf{b}$, the count of specific integer
solutions to the system of linear equations $A\mathbf{x} = \mathbf{b}$ with the
entries of $\mathbf{x}$ in $[n]_p$ follows a (conveniently rescaled) normal
limiting distribution. This applies among others to the number of solutions
with every variable having a different value, as well as to a broader class of
so-called non-trivial solutions in homogeneous strictly balanced systems. Our
proof relies on the delicate linear algebraic study both of the subjacent
matrices and the corresponding ranks of certain submatrices, together with the
application of the method of moments in probability theory.
