Sum of all integer binary solutions of a TUM linear system

Question

I have the following problem: $A x = b$ where $A$ is a $m \times n$ total unimodular matrix (TUM) with entries in $\{0,1\}$ and $b$ is a $m$-vector of strictly positive integers. Let $\mathcal X$ be the set of all the $0-1$ solutions to the linear system. I need to find the sum (or the mean) of all the element in $\mathcal X$, in other words the vector $\sum_{x \in \mathcal X} x$ or the vector $\frac{1}{|\mathcal X|}\sum_{x \in \mathcal X} x$, possibly without explicitly enumerating all solutions (using backtrack methods, for example).