The probability of successful allocation of particles in cells (the general case)

Let $p_{nN}$ be the probability of successful allocation of $n$ groups of particles in $N$ cells with the following assumptions: (a) each group contains $m$ particles and has allocation as a general allocation scheme; (b) each cell contains at most $r$ particles from the same group; (c) events connected with different groups are independent. We obtain an asymptotically exact bound of $p_{nN}$ as $n,N\to\infty$ such that $n/N$ is bounded. Applications to problems in error-correcting coding are considered.