(** The falling factorial *)
Definition falling_factorial := nosimpl ffact_rec.
Notation "n ^_ m" := (falling_factorial n m)
(at level 30, right associativity) : nat_scope.

(* In fact the only exception for bin_factd is n = 0 and m = 1 *)
Lemma bin_factd n m : 0 < n -> 'C(n, m) = n`! %/ (m`! * (n - m)`!).