  Let $\rm{Mod(S)}$ be the mapping class group of a closed orientable surface
$S$ of genus $g \geq 2$. Let $G$ be a non-elementary subgroup of $\rm{Mod(S)}$
so that the associated Bowen-Margulis measure is finite. In this paper, we give
an asymptotic growth formula for $G$ with respect to the Teichm\"{u}ller
metric.
