  Nucleosynthesis calculations require nuclear level densities for hundreds or
even thousands of nuclides. Ideally one would like to constrain these level
densities by microscopically motivated yet computationally cheap models. A
statistical approach suggests that low moments of the Hamiltonian might be
sufficient. Recently Zuker proposed a simple combinatoric formula for level
densities based upon the binomial distribution. We rigorously test the binomial
formula against full scale shell-model diagonalization calculations for
selected $sd$- and $pf$-shell nuclides and also against Monte Carlo path
integration calculations. We find that the fourth moment is as important as the
third moment to a good description of the level density, as well as
partitioning of the model space into subspaces.
