On the semigroup of transformations of a finite set generated by random generators

We consider the semigroup generated by random mappings and random bijective mappings of a finite set $\Omega_n$ of cardinality $n$ into itself. We study the question when this semigroup includes all mappings of $\Omega_n$ into itself with a fixed cardinality $k$ of the image of the set $\Omega_n$. As $n\to\infty$, the ranges of $k$ are given where this inclusion holds with probability tending to zero or one, and two domains of values of $k$ where the inclusion holds with intermediate probability.