  We report on a simple strategy to treat mean-field limits of quantum
mechanical systems in which a large number of particles weakly couple to a
second-quantized radiation field. Extending the method of counting, introduced
in [Lett. Math. Phys. 97, 151-164], with ideas inspired by
[http://www.mathematik.uni-muenchen.de/%7Ebohmmech/theses/Matulevicius_Vytautas_MA.pdf]
and [J. Math. Phys. 54(1), 012303] leads to a technique that can be seen as a
combination of the method of counting and the coherent state approach. It is
similar to the coherent state approach but might be slightly better suited to
systems in which a fixed number of particles couple to radiation. The strategy
is effective and provides explicit error bounds. As an instructional example we
derive the Schr\"odinger-Klein-Gordon system of equations from the Nelson model
with ultraviolet cutoff. Furthermore, we derive explicit bounds on the rate of
convergence of the one-particle reduced density matrix of the non-relativistic
particles in Sobolev norm. More complicated models like the Pauli-Fierz
Hamiltonian can be treated in a similar manner [arXiv:1609.01545].
