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How can I prove from first principles that $0!$ is equal to $1$?

  • 16
    For any cardinal c let c! be the order of the group of permutations of a set of cardinality c.2011-02-08
  • 33
    You haven't stated what your definition of factorial is. An inductive definition would have as the base case 0! = 1, so there's nothing to prove from that definition, for example.2011-02-08
  • 55
    Please don't state questions as orders; write them as questions.2011-02-09
  • 5
    Or to put it differently, $0!=1$ *is* one of the "first principles" in the most typical formulation.2011-09-27
  • 21
    Wow, I've been coding too much lately and read this question as asking to show `0 != 1`, (or for the non coders $0\neq 1$).2012-10-03
  • 2
    What do you consider as "first principles"?2012-11-17
  • 4
    Because there is ONLY ONE way to do nothing. http://math.stackexchange.com/questions/20969/prove-0-1-from-first-principles/485421#4854212014-01-13
  • 0
    Simple. Instead of defining $0!$, define $1!$ instead, then prove that $0! = 1$.2016-06-24

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