Particle occupation factors without large number approximations

New derivations of particle occupation factors that are based on mean values and do not require large number approximations (LNA) are provided for fermions, bosons, and Boltzmann particles. The derivations are closely related to traditional combinatorial approaches, so the physical content of the latter is preserved without recourse to most probable values or LNA. The approach is based on the use of eigenvalues of the density matrix as probabilities entering into the Shannon entropy. The degeneracies of the eigenvalues of the density matrix can be identified with Boltzmann enumeration factors. Problems associated with steepest descent procedures, limit theorem techniques, and the Boltzmann thermodynamic probability are avoided. In addition, since the approach is based on the eigenvalues of the density matrix, the concept of ensembles is not required.