Limit theorems for the number of empty cells

We consider the scheme of equiprobable allocating $n$ groups of particles, generally speaking, of different volumes, to $N$ cells. For $N\to\infty$ and various relations between the parameters of the scheme, we investigate the asymptotic behaviour of the number of empty cells among the $M$ chosen cells after allocating $n$ groups of particles. We obtain an integral and a local limit theorems on convergence to the normal law, as well as the Poisson-like limit theorems. In all cases considered, an estimate of the convergence rate is given.