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We consider the set Sigma(R,C) of all mxn matrices having 0-1 entries andprescribed row sums R=(r_1, ..., r_m) and column sums C=(c_1, ..., c_n). Weprove an asymptotic estimate for the cardinality |Sigma(R, C)| via the solutionto a convex optimization problem. We show that if Sigma(R, C) is sufficientlylarge, then a random matrix D in Sigma(R, C) sampled from the uniformprobability measure in Sigma(R,C) with high probability is close to aparticular matrix Z=Z(R,C) that maximizes the sum of entropies of entries amongall matrices with row sums R, column sums C and entries between 0 and 1.Similar results are obtained for 0-1 matrices with prescribed row and columnsums and assigned zeros in some positions.

On the number of matrices and a random matrix with prescribed row and column sums and 0-1 entries