  Based on the non-relativistic QCD factorization formalism, we calculate the
bottomonium ground state, $\eta_b$, inclusive charm decays at the leading order
in the strong coupling constant $\alpha_s$ and quarkonium internal relative
velocity $v$. The inclusive charm pair production in $\eta_b$ decay is mainly
realized through $\eta_b \to c \bar{c} g$ process, where the charm and
anti-charm quarks then dominantly hadronize into charmed hadrons. The momentum
distributions of the final states are presented. In this work, we also
calculate the $J/\psi$ inclusive production rate in the $\eta_b$ decays, where
the color-octet contribution is found to be very important. We expect this
study may shed some light on finding $\eta_b$ or knowing more about its nature.
