  We prove an asymptotic estimate for the number of mxn non-negative integer
matrices (contingency tables) with prescribed row and column sums and, more
generally, for the number of integer feasible flows in a network. Similarly, we
estimate the volume of the polytope of mxn non-negative real matrices with
prescribed row and column sums. Our estimates are solutions of convex
optimization problems and hence can be computed efficiently. As a corollary, we
show that if row sums R=(r_1, ..., r_m) and column sums C=(c_1, ..., c_n) with
r_1 + ... + r_m =c_1 + ... +c_n =N are sufficiently far from constant vectors,
then, asymptotically, in the uniform probability space of the mxn non-negative
integer matrices with the total sum N of entries, the event consisting of the
matrices with row sums R and the event consisting of the matrices with column
sums C are positively correlated.
