  Let $S_{g,n}$ be an oriented surface of genus $g$ with $n$ punctures, where
$2g-2+n>0$ and $n>0$. Any ideal triangulation of $S_{g,n}$ induces a global
parametrization of the Teichm\"uller space $\mathcal{T}_{g,n}$ called the
shearing coordinates. We study the asymptotics of the number of the mapping
class group orbits with respect to the standard Euclidean norm of the shearing
coordinates. The result is based on the works of Mirzakhani.
