The count for ways of doing each way of matching the bins was (popcount - popbinheight) ^ (sampcount - sampbinheight).
We also figured some things possibly around permutations and combinations, both.
If you are looking for how many ways to make sampbinheight bins from only a popbinheight bin, and selecting one in a different decision is a different event, then the answer could be (popbinheight P sampbinheight) .... Oh no I think this is wrong! Permutations assume they go away when you pick them, but the distribution does not change when sampled from.
And remember to simulate it to find more and more mistakes.
When we were sampling from decisions, there were only so many, but from a population summarised by a histogram, there are enough to use replacement. Is this permutations with replacement? Is that even a thing?