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In mathematics, the factorial of a non-negative integer ''n'', denoted by ''n''!, is the product of all positive integers less than or equal to ''n''. For example, : The value of 0! is 1, according to the convention for an empty product.〔Ronald L. Graham, Donald E. Knuth, Oren Patashnik (1988) ''Concrete Mathematics'', Addison-Wesley, Reading MA. ISBN 0-201-14236-8, p. 111〕 The factorial operation is encountered in many areas of mathematics, notably in combinatorics, algebra, and mathematical analysis. Its most basic occurrence is the fact that there are ''n''! ways to arrange ''n'' distinct objects into a sequence (i.e., permutations of the set of objects). This fact was known at least as early as the 12th century, to Indian scholars.〔N. L. Biggs, ''The roots of combinatorics'', Historia Math. 6 (1979) 109−136〕 Fabian Stedman in 1677 described factorials as applied to change ringing.〔 The publisher is given as "W.S." who may have been William Smith, possibly acting as agent for the Society of College Youths, to which society the "Dedicatory" is addressed.〕 After describing a recursive approach, Stedman gives a statement of a factorial (using the language of the original):
The notation ''n'' The definition of the factorial function can also be extended to non-integer arguments, while retaining its most important properties; this involves more advanced mathematics, notably techniques from mathematical analysis. ==Definition== The factorial function is formally defined by the product : or by the recurrence relation : The factorial function can also be defined by using the power rule as :〔http://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2006/lecture-notes/lec4.pdf〕 All 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 , 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「factorial」の詳細全文を読む スポンサード リンク
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