The probability of a successful allocation of ball groups by boxes

Let $p=p_{Nn}$ be the probability of a successful allocation of $n$ groups of distinguishable balls in $N$ boxes in equiprobable scheme for group allocation of balls with the following assumption: each group contains $m$ balls and each box contains not more than $q$ balls from a same group. If $q=1$, then we easily calculate $p$ and observe that $p\to e^{-\frac{m(m-1)}2\alpha_0}$ as $n,N\to\infty$ such that $\alpha=\frac nN\to\alpha_0<\infty$. In the case $2\le q$ we also find an explicit formula for the probability and prove that $p\to1$ as $n,N\to\infty$ such that $\alpha=\frac nN\le\alpha'<\infty$.