if your calculations over the factorial numbers are only multiplications and divisions (as I think they are), you can operate over their log values instead, transforming multiplications and divisions to additions and substractions respectively. i.e.:

log ($n! / ($r! * ($n-$r)!)
  = Sum(log(1)..log($n))
    - Sum(log(1)..log($r))
      - Sum(log(1)..log($n-$r))
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and as your operations will only involve a small number of integers, you can cache log($n) and log($n!) to speed up the calculations.