Mathematicians know the "hat check" problem: If N people get hats back at random, what are the odds that no one gets their own hat back? It's the nearest fracti

Mathematicians know the "hat check" problem: If N people get hats back at random, what are the odds that no one gets their own hat back? It's the nearest fraction with N! denominator to 1/e, a beautiful counting problem. Very close to 1/e (off by about 1/6!) even for _FIVE_ people. I too wonder if 1,000 is off the mark.