The distribution of the number of different elements of a symmetric basis in a random $mA$-sample

A general combinatorial model is studied in terms of which, for example, the problem of disposal of $m$ different objects into $n$ identical cells or the problem of partitions of a set consisting of $m$ elements into disjoint subsets could be discribed.
It is proved, in particular, that, under some conditions laid on a subsequence $A$ of positive integers, the number of subsets with the powers in $A$ of a divided at random set consisting of $m$ elements is asymptotically normal as $m\to\infty$.