Sorry for posting more, but my cognition is so contextual, and
   obviously that answer is wrong.
   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?