On the distribution of the number of solutions of random systems of equations which are known to be consistent

The distribution of the number of solutions of the systems in which each equation is specified by the substitution into a function $\varphi(u_1,\dots,u_d)$, $u_j\in\{0,1\}$, binary unknowns taken at random and without replacement from the set $\{x_1,\dots,x_n\}$, $n\ge d$, is studied. It is proved that, under certain conditions the distribution of the logarithm to base 2 of the number of solutions of the obtained system converges to a Poisson distribution.