Generalized allocation scheme with cell occupancies from a fixed finite set

We consider a generalized scheme of allocation of $n$ particles (elements) over unordered cells (components) under the condition that the number of particles in each cell belongs to a fixed finite set $A$ of positive integers. A new asymptotic estimates for the total number $I_n(A)$ of variants of allocations of $n$ particles are obtained under some conditions on the set $A$; these estimates have an explicit form (up to equivalence). Some examples of combinatorial-probabilistic character are given to illustrate by particular cases the notions introduced and results obtained. For previously known theorems on the convergence to the normal law of the total number of components and numbers of components with given cardinalities the norming parameters are obtained in the explicit form without using roots of algebraic or transcendent equations.