Combinatorial scheme of random placement of particles into cells of several types

Based on the $A$-scheme of sequential trials, we examine two variants of random placement of particles in cells of $r$ types. In one of these variants, there exist marked cells of each type. We find an explicit distribution of the number of nonempty cells in one of the variants and a distribution of the number of nonempty marked cells in the other, obtain numerical characteristics of these distributions, and prove limit theorems.