The necessary and sufficient conditions are found for convergences of series of weighted probabilities of large deviations for combinatorial sums ∑iXniπn(i), where ||Xnij|| is an n-order matrix of independent random variables and (πn(1), πn(2), …, πn(n)) is a random permutation with the uniform distribution on the set of permutations of numbers 1, 2, …, n independent of Xnij. Combinatorial variants of the results of convergence rates are obtained in the strong law of large numbers and in the law of the iterated logarithm under close to optimal conditions. Applications to rank statistics are discussed.