  In this work, we generalize M. Kac's original many-particle binary stochastic
model to derive a space homogeneous Boltzmann equation that includes a linear
combination of higher-order collisional terms. First, we prove an abstract
theorem about convergence from a finite hierarchy to an infinite hierarchy of
coupled equations. We apply this convergence theorem on hierarchies for
marginals corresponding to the generalized Kac model mentioned above. As a
corollary, we prove propagation of chaos for the marginals associated to the
generalized Kac model. In particular, the first marginal converges towards the
solution of a Boltzmann equation including interactions up to a finite order,
and whose collision kernel is of Maxwell-type with cut-off.
