  Let $\phi$ be an even Hecke-Maass cusp form on ${\rm SL}_2(\mathbb{Z})$ whose
$L$-function does not vanish at the center of the functional equation. In this
article, we obtain an exact formula of the average of triple products of
$\phi$, $f$ and $\bar f$, where $f$ runs over an orthonormal basis $H_k$ of
Hecke eigen elliptic cusp forms on ${\rm SL}_2(\mathbb{Z})$ of a fixed weight
$k\geq 4$. As an application, we prove a quantitative non-vanishing results on
the central values for the family of degree $6$ $L$-functions $L(s,\phi \times
{\rm Ad}\,f)$ with $f$ in the union of $H_k$ $({\rm K} \leq k < 2{\rm K})$ as
${\rm K}\rightarrow \infty$.
