Asymptotic behaviour of a number of groups of particles in a classical problem of permutation

Let groups each of $m$ particles be distributed independently into $n$ cells so that particles of every group are distributed into different cells with all ${n\choose m}$ possible permutations having equal probabilities. A random variable $\nu_m(n,t)$ is introduced which is equal to the number of groups whose distribution leads to at least $t$ cells being occupied for the first time.
In this paper the whole spectrum of limit theorems is obtained and exact formulae as well as their asymptotic expressions as $n$, $t\to\infty$ of the mean and variance of random variables $\nu_m(n,t)$ are found.