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Three- and four-cluster breakup reactions in the ${}^{16}$C-proton scatteringare studied using three-body core-neutron-neutron model for ${}^{16}$C.Single-scattering approximation (SSA) of four-particle equations for transitionoperators is used to calculate three- and four-cluster breakup amplitudes at200 and 300 MeV/nucleon energy near proton-neutron ($pn$) quasi free scattering(QFS) conditions. The differential cross section is sharply peaked at $pn$ QFSpoint and decreases rapidly whenever kinematical conditions deviate from $pn$QFS. The accuracy of the SSA for the three-cluster breakup is estimated from athree-body model and is found to be as good as 6\% at higher reaction energiesin suitable angular regimes. Furthermore, under an additional approximation thethree-cluster breakup amplitude factorizes into the $pn$ transition operatorand the overlap integral of two- and three-particle bound states. Thatapproximation usually reduces the cross section, in some cases even up to 10\%.

Breakup of a nucleus with two weakly bound neutrons on a proton target: Single-scattering approximation