Limit theorems in a model of the arrangement of paticles of two types

Particles of two types are thrown independently into $N$ cells. A particle of the kth type gets into the ith cell with a probability $a_i^k$, $k=1,2$, $i=1,\dots,N$. Denote by $\mu_0^{(k)}(n_k)$ the number of cells which contain no particles of type $k$ ($k=1,2$) and by $\mu_0^3(n_1+n_2)$ the number of cells which contain no particles at all. In this paper some limit theorems for $\mu_0^{(1)}(n_1)$, $\mu_0^{(2)}(n_2)$ and $\mu_0^{(3)}(n_1+n_2)$ are proved.