I need help with an approximation concerning the birthday problem. In a recent MAA *Monthly* (August-September 2013) article "Simple Approximation Formulas for the Birthday Problem" by Matthias Arnold and Werner Glass, the authors claim the probability $p$ that out of $n$ people at least $k$ birthdays occur on a single day in a year with $c$ days is approximately
$$p\approx1-\exp\,\left(\!n^k{\textrm{e}}^{-n/c}\left(\!\frac n{c(k+1)} -1\!\right)^{\!-1}c^{1-k}(k!)^{-1}\!\right).$$

The authors cite this formula from a 1989 paper "Methods for Studying Coincidences" by Persi Diaconis and Frederick Mosteller. Unfortunately, this reference only claims the formula comes from unpublished work and provides no further references. Can anyone provide insight about how to obtain such a formula?