Is there or is there not a reasonable way to measure this?  What would a stats teacher tell you?

Enumerating all the samplings involves considering each population bin, with each next population bin, and each next population bin.  This pattern is why the total number of possible histogram samplings is popcount^sampcount or some such.

If this bin has a height of z, that means that exactly z of those iterations landed on it, and sampcount-z of them did not.

The sampcount-z quantity is a product that will be the same for every way of filling the bin.  So the big question is, how many ways are there of filling the bin.