  Let $G$ be a permutation group acting on a finite set $\Omega$ of cardinality
$n$. The number of orbits of the induced action of $G$ on the set $\Omega_m$ of
all size $m$ subsets of $\Omega$ satisfies the trivial inequalities
$|\Omega_m|/|G|\leq |\Omega_m/G|\leq |\Omega_m|$. The paper offers improvements
of the upper bound in terms of the minimal degree of $G$ or the minimal degree
of some its subset with a small complement. Applications include asymptotic
enumeration of point configurations in an affine space over a finite field,
unlabeled graphs and hypergraphs. Finally, with references to known results of
permutation groups theory it is shown that if $G$ is an arbitrary 2-transitive
group except for $S_n$ and $A_n$, then $|\Omega_m/G|\approx |\Omega_m|/|G|$ for
$m$ and $n$ large provided the ratio $m/n$ is bounded away from 0 and 1.
  Similar results hold for the induced action of $G$ on the set $\Omega_{(m)}$
of all weight $m$ multisets on $\Omega$ provided the ratio $m/n$ is not too
small.
