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