Two-proton and three-proton inclusive ${\mathrm{\ensuremath{\pi}}}^{+}$ absorption cross sections were measured for incident pion energies from 250 to 500 MeV on targets of $^{12}\mathrm{C}$, $^{58}\mathrm{Ni}$, $^{90}\mathrm{Zr}$, $^{118}\mathrm{Sn}$, and $^{208}\mathrm{Pb}$ using a large solid angle bismuth germanate detector array. Pion absorption events leading to three or more protons in the final state were observed to become more significant with increasing incident pion energy and were dominant at 500 MeV. The total absorption cross sections were inferred from analyses of the data with uncertanties of approximately 25%.