Volume of convex polytope the vertices of which are vertices of the unit hypercube

Question

I have an infinite family $P_{a,b}$ of non-degenerate convex polytopes of dimension $ab$. Each polytope is given explicitely by a list of its vertices and all of these vertices are elements of $\{0,1\}^{ab}$, i.e. vertices of the unit hypercube. Thus there exists a decomposition into simplices of volume $\frac{1}{(ab)!}$ and thus the volume of $P_{a,b}$ is an integer multiple of $\frac{1}{(ab)!}$. In general it is very hard to compute volumes of convex polytopes of high dimension but does somebody happen to know any kind of explicit formula in this special situation (or maybe a closely related one)?