Number of Integer Solutions Problem An elevator in the Empire State Building starts at the basement with 57 people (not including the elevator operator) and discharges them all by the time it reaches the 86th floor. In how many ways could the operator have perceived the people leaving the elevator if they all looked different to him?

I have another result. Start with a small example. You have 2 distinguishable urns (floors) and 3 numbered balls (people).
The possible ways putting the balls into the urns:
$\text{urn 1} \ | \ \text{urn 2}$
$\text{1,2} \ \ \ | \ \ \ \ \text{3}$
$\text{1,3} \ \ \ | \ \ \ \ \text{2}$
$\text{3,2} \ \ \ | \ \ \ \ \text{1}$
$\ \text{1} \ \ \ \ | \ \ \ \ \text{2,3}$
$ \ \text{2} \ \ \ \ | \ \ \ \ \text{1,3}$
$ \ \text{3} \ \ \ \ | \ \ \ \ \text{1,2}$
$ \ \text{-} \ \ \ \ | \ \ \ \ \text{1,2,3}$
$\text{1,2,3} \ \ \ \ | \ \ \ \ \text{-}$
Thus there are $2^3=8$ ways. In your case the are $86^{57}$ ways.
