(popbinheight^sampbinheight) (popcount - popbinheight) ^ (sampcount - sampbinheight) (sampcount C sampbinheight)
Is the combinations still looking ok, now that the system is describsd differently?
The question is whether or not the expression counts all the possible different decision sets, without duplication, that result in exactly sampbinheight samples from popbinheight population portions.
Goal: validate expression. Note: expression is useful on computer, not mailing list.
We can break the outcomes into sets based on which of the decisions are the ones that pick the bin of interest. This is a selection of sampbinheight decisions from among sampcount total decisions, which cannot be reordered. (sampcount C sampbinheight).
Each of these sets has the same two unknown spaces, none of which overlap, of which non-relevent decisions are made, and which of the popbinheight samples are selected. The non-relevent decisions are made from a nondepleting group that is (popcount - popbinheight) large. The sample choices are each made from a homogenous group that is popbinheight large. Selecting a different item from popbinheight is a different decision, the same way selecting a different nonrelevent item is ... I think?
So we have (popcount - popbinheight) ^ (sampcount - sampbinheight) and we have popbinheight ^ sampbinheight .
These are the same expressions listed above.
I'm sure I made another error, but I don't know what it is, and it's nice to have a better guess.
Remember to simulate to find more mistakes.