  We explore the superconducting pairing mechanisms in the trilayer
$\mathrm{La}_4\mathrm{Ni}_3\mathrm{O}_{10}$ material through self-consistent
mean-field calculations. Our findings demonstrate that intralayer pairings are
substantially weaker compared to interlayer ones. Remarkably, in the state
characterized by interlayer pairing, we detect the presence of partially gapped
Fermi surfaces, a fascinating occurrence attributable to the disparity between
the inner and outer conducting layers of
$\mathrm{La}_4\mathrm{Ni}_3\mathrm{O}_{10}$. Moreover, this study provides
valuable insights into the lower superconducting transition temperatures
observed in $\mathrm{La}_4\mathrm{Ni}_3\mathrm{O}_{10}$ compounds. This
contributes to a deeper understanding of its distinct superconducting
attributes.
