The wrong part I found is that I am assuming binheight specific samples need to be chosen, when really eaxctly sampbinheight samples need to be chosen from a popbinheight constant set among popcount.

The sampcount - sampbinheight samples that are not involved are still the similar (popcount - popbinheight) ^ (sampcount - sampbinheight) total outcomes.

Then we have popbinheight many population samples that are selected exactly sampbinheight times from sampcount selections.

Maybe the combinations can give us how many different decisions it can occur in, and then maybe 

Noting that simulating this is important because it is so hard for me to verify logical things.

If combinations are the factor, then the next queation would be how many different ways can we select sampbinheight items from popbinheight items.  Here, do we want the count of ordered lists, or the count of unordered sets?