If *n* is a positive integer, then its *factorial,* written *n*!, is the number 1 × 2 × · · · × *n*: that is, the product of all positive integers up to *n*. For example, the first eight factorials are 1, 2, 6, 24, 120, 720, 5040, and 40320. (The exclamation mark was introduced by Christian Kramp 200 years ago as a convenience to the printer: it is perhaps also intended to convey some alarm at the rapidity with which *n*! grows. An obsolete notation, which can still be found in some twentieth-century texts, is .) From this definition, it might appear to be impossible to make sense of the idea of the factorial of a number ...

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