The Bird and Nanbu systems are particle systems used to approximate the solution of the mollified Boltzmann equation. In particular, they have the propagation of chaos property. Following [GM94], we use coupling techniques and results on branching processes to write an expansion of the error in the propagation of chaos in terms of the number of particles, for slightly more general systems than the ones cited above. This result leads to the proof of the a.s convergence and the central-limit theorem for these systems. In particular, we have a central-limit theorem for the empirical measure of the system at a fixed time t under far less stringent assumptions then in [Mel98].