The authors use probabilistic arguments in the spirit of the De Moivre-Laplace central limit theorem to obtain asymptotic estimates for combinatorial sums. The main question under discussion is the following. For a given sequence \(\chi_n=\sum_{\lambda\vdash n}f(\lambda)\chi_{\lambda}\) of \(S_n\)-characters defined in terms of the associated Young diagrams and with known asymptotics of the degrees when \(n\) tends to infinity, increase the length of the columns of the Young diagrams by a fixed factor \(q\), i.e. replace the partitions \(\lambda=(\lambda_1,\ldots,\lambda_k)\) with the new partitions \(q\ast\lambda= (\lambda_1^q,\ldots,\lambda_k^q)= (\lambda_1,\ldots,\lambda_1,\ldots, \lambda_k,\ldots,\lambda_k)\). Then determine the changes that occur in the asymptotics of the degrees of the new sequence \(\chi_{qn}= \sum_{\lambda\vdash n}f(\lambda)\chi_{q\ast\lambda}\) of \(S_{qn}\)-characters. The authors determine such estimates for the so-called Young derived sequences of characters. As a result they obtain some rather interesting integration formulas. In particular, some of the integrals involved are extensions of Selberg and Dyson-Macdonald-Mehta type integrals.