Factorial - Definition

Definition

The factorial function is formally defined by

or recursively defined by

n! = \begin{cases} 1 & \text{if } n = 0, \\ (n-1)!\times n & \text{if } n > 0. \end{cases}

Both of the above definitions incorporate the instance

in the first case by the convention that the product of no numbers at all is 1. This is convenient because:

  • There is exactly one permutation of zero objects (with nothing to permute, "everything" is left in place).
  • The recurrence relation (n + 1)! = n! × (n + 1), valid for n > 0, extends to n = 0.
  • It allows for the expression of many formulae, such as the exponential function, as a power series:
  • It makes many identities in combinatorics valid for all applicable sizes. The number of ways to choose 0 elements from the empty set is . More generally, the number of ways to choose (all) n elements among a set of n is .

The factorial function can also be defined for non-integer values using more advanced mathematics, detailed in the section below. This more generalized definition is used by advanced calculators and mathematical software such as Maple or Mathematica.

Read more about this topic:  Factorial

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