  Inclusive S- and P-wave charmonium productions in the bottomonium ground
state $\eta_b$ decay are calculated at the leading order in the strong coupling
constant $\alpha_s$ and quarkonium internal relative velocity $v$ in the
framework of the NRQCD factorization approach. We find the contribution of
$\eta_b \to \chi_{c_J}+ g g$ followed by $\chi_{c_J} \to J/\psi+ \gamma$ is
also very important to the inclusive $J/\psi$ production in the $\eta_b$
decays, which maybe helpful to the investigation of the color-octet mechanism
in the inclusive $J/\psi$ production in the $\eta_b$ decays in the forthcoming
LHCb and SuperB. As a complementary work, we also study the inclusive
production of $\eta_c$, and $\chi_{cJ}$ in the $\eta_b$ decays, which may help
us understand the X(3940) and X(3872) states.
