How Joe Gillis Discovered Combinatorial Special Function Theory

There are EXACTLY 1493804444499093354916284290188948031229880469556 Ways to Derange a Standard Deck of Cards (ignoring suits) [and many other such useful facts]

In this memorial tribute to Joe Gillis, who taught us that Special Functions count, we show how the seminal Even-Gillis integral formula for the number of derangements of a multiset, in terms of Laguerre polynomials, can be used to efficiently compute not only the number of the title, but much harder ones, when it is interfaced with Wilf-Zeilberger algorithmic proof theory.