  Recently, evidence of superconductivity (SC) has been reported in pressurized
La$_4$Ni$_3$O$_{10}$. Here we study the possible pairing mechanism and pairing
symmetry in this material. Through fitting the density-functional-theory band
structure, we provide a six-orbital tight-binding model. In comparison with the
band structure of La$_3$Ni$_2$O$_7$, the additional non-bonding $d_{z^2}$ band
is importance to the pairing mechanism here. When the multi-orbital Hubbard
interactions are included, our random-phase-approximation based study yields an
$s^{\pm}$-wave pairing. The dominant FS nesting with nesting vector
$\mathbf{Q}_1\approx (\pi,\pi)$ is between the $\gamma$-pocket contributed by
the bonding $d_{z^2}$ band top and the $\alpha_1$-pocket contributed by the
non-bonding $d_{z^2}$ band bottom, leading to the strongest pairing gap
amplitude and opposite gap signs within the two regimes. The dominant
real-space pairing is the interlayer pairing between the $d_{z^2}$ orbitals. We
have also studied the doping dependence of the pairing symmetry and $T_c$.
