On the probability of the event: in $n$ generalized allocation schemes the volume of each cell does not exceed $r$

We consider $n$ identical generalized schemes of allocating particles in cells. We study the probability of the event: for each generalized allocation scheme, there are at most $r$ particles in each cell, where $r$ is a given number. We obtain an asymptotic estimate for this probability and we consider the application of the obtained results to an antinoise coding.