  The LHC$_b$ collaboration has systematically measured the rates of $B_c\to
J/\psi K$, $B_c\to J/\psi D_s$, $B_c\to J/\psi D_s^*$ and $B_c\to \psi(2S)
\pi$. The new data enable us to study relevant theoretical models and further
determine the model parameters. In this work, We calculate the form factors for
the transitions $B_c\to J/\psi$ and $B_c\to \psi(2S)$ numerically, then
determine the partial widths of the semi-leptonic and non-leptonic decays. The
theoretical predictions on the ratios of $\Gamma(B_c\to J/\psi K)/\Gamma(B_c\to
J/\psi \pi)$, $\Gamma(B_c\to J/\psi D_s)/\Gamma(B_c\to J/\psi \pi)$ and
$\Gamma(B_c\to J/\psi D_s^*)/\Gamma(B_c\to J/\psi \pi)$ are consistent with
data within only 1$\sigma$. Especially, for calculating $\Gamma(B_c\to
\psi(2S)X)$ the modified harmonic oscillator wave function (HOWF) which we
developed in early works is employed and the results indicate that the modified
harmonic oscillator wave function works better than the traditional HOWF.
