We report a systematic study of the normal-state and superconducting properties of single-crystal ${\mathrm{Pb}}_{1-\mathrm{x}}{\mathrm{Sn}}_{\mathrm{x}}{\mathrm{TaSe}}_2$ $(0\le x\le 0.23)$. Sn doping enhances the superconducting temperature $T_c$ up to 5.1 K while also significantly increasing impurity scattering in the crystals. For $x=0$ and 0.018, the specific-heat jump at $T_c$ exceeds the Bardeen-Cooper-Schrieffer (BCS) weak-coupling value of 1.43, indicating the realization of strong-coupling superconductivity in undoped and slightly-Sn-doped ${\mathrm{PbTaSe}}_2$. Substituting Pb with more Sn lowers the specific-heat jump at $T_c$ below the BCS value of 1.43, which cannot be explained by a single-gap model. Rather, the observed specific-heat data of moderately-Sn-doped ${\mathrm{PbTaSe}}_2$ $(x=0.08 \text{and} 0.15)$ are reproduced by a two-gap model. Our density functional theory calculations suggest that three-dimensional Fermi pockets appear due to a reduction of the spin-orbit gap with Sn doping, and the multiband effect arising from these emergent Fermi pockets enhances the effective electron-phonon coupling strength, leading to the increase in $T_c$ of ${\mathrm{Pb}}_{1-\mathrm{x}}{\mathrm{Sn}}_{\mathrm{x}}{\mathrm{TaSe}}_2$.

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• Multiband superconductivity
• Specific heat
• Spin-orbit coupling
• Nodal-line semimetals
• Calorimetry
• Resistivity measurements