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In Mathematical, the factorial of a Whole naturalness N , noted N ! , which bed either “factorial of N” or “factorial N”, is the produced of the integers strictly positive inferiors or equal to N .

The factorial plays a big role in combinative algebra because there is N ! different ways of permute N objects. It appears in many formulas in mathematics, such as for example the formula of the binomial and the Formule of Taylor.

Definition

That is to say $n$ a natural entirety. Its factorial is formally defined by:

$n!\; =\; \backslash \; prod\_\; \{i=1\}\; ^n\; I\; =\; 1\; \backslash \; times\; 2\; \backslash \; times\; 3\; \backslash \; times\; \backslash \; cdots\; \backslash \; times\; (n-1)\; \backslash \; times\; n$

For example:

• 1! = 1
• 2! = 1 × 2 = 2
• 3! = 1 × 2 × 3 = 6
• 10! = 1 × 2 × 3 × 4 × 5 × 6 × 7 × 8 × 9 × 10 = 3.628.800

By convention:

$0!\; =\; 1$

The definition of the factorial in the form of product makes natural this convention since 0! is a Produit vacuum, i.e. reduced to the neutral element of the multiplication. This convention is practical for two points:

• It allows a recursive definition factorial: (n+1)! = N! × (n+1) for all N .
• It allows the many ones identifés into combinative to be valid for null sizes. In particular, the number of arrangements or permutations of the Empty set is equal to 1.

The Formule of Stirling gives an equivalent of N ! when N is large:

$\backslash \; lim\_\; \{N\; \backslash \; to+\; \backslash \; infty\}\; \backslash \; frac\; \{N!\}\{\backslash \; sqrt\; \{2\; \backslash \; pi\; N\}\; (n/e)\; ^n\}\; =1$

from where $n\; \backslash !\; \backslash \; sim\; \backslash \; sqrt\; \{2\; \backslash \; pi\; N\}\; \backslash ,\; \{\backslash \; left\; (\backslash \; frac\; N\; E\; \backslash \; right)\}^n$

Generalization

See also: Function Gamma of Euler

The factorial function can be prolonged with the whole of the complex numbers (except for the negative or null integers) thanks to the function gamma of Euler (noted Γ). Indeed, for positive N whole, one a:

$\backslash \; Gamma\; (n+1)\; =n!$

In addition, the two functions satisfy the following relations of recurrence:

$n!=n\; \backslash \; times\; (n-1)!$

$\backslash \; Gamma\; (n+1)\; =n\; \backslash \; Gamma\; (N)$

The function gamma thus acts like a prolongation of the factorial:

$z!\; =\; \backslash \; Gamma\; (z+1)\; =\; \backslash \; int\_\; \{0\}\; ^\; \{\backslash \; infty\}\; t^z\; e^\; \{-\; T\}\; \backslash ,\; \backslash \; mathrm\; \{D\}\; t$

This function is however not defined for the negative or null integers (0, -1,-2, etc).

This vision of the function gamma as prolongation of the factorial is justified by the following reasons:

• the two functions share the same recurring definition.
• the function gamma is generally used in a similar context (even if more general) with the factorial.
• the function gamma is the only function which satisfies this definition of recurrence on the complex numbers, which is holomorphic and whose Logarithme of the restriction on positive realities is convex (Théorème of Bohr-Mollerup)

Applications

• In Combinative, there exists N ! different ways to arrange N distinct objects (i.e. N ! Permutation S). And the number of ways of choosing K elements among a whole of N is given by the binomial Coefficient:

$\{N\; \backslash \; choose\; K\}\; =\; \{N!\; \backslash \; over\; K!\; (n-k)!\}.$

• In Permutation, if R elements can be selected and arranged R ways different among a total from N objects ( R < N ), then the full number of distinct permutations is given by:

_$n$P_$r$ = $\backslash \; frac\; \{N!\}\{(NR)!\}$

• the factorials also appear in analyzes. For example, the Théorème of Taylor, which expresses the value in X of a function F in the form of whole Série, utilizes the factorial N ! for the term corresponding to the Derived from F in X .

• the Volume of a Hypersphère out of N dimensions can be expressed by:

$V\_n=\; \{\backslash \; pi^\; \{n/2\}\; R^n\; \backslash \; over\; (n/2)!\}.$

• the factorials are used in an intensive way in Theory of probability.

• the factorials are often used as example - with the Suite of Fibonacci - for the training of the Récursivité in Informatique because of their simple recurring definition.

Theory of the numbers

The factorials have many applications in Théorie of the numbers. The factorial numbers are highly made up numbers. In particular, N ! by all the prime numbers is divisible which is equal or lower to him. Consequently, any number N > 4 is a made up Nombre if and only if:

$(n-1)!\; \backslash \; \backslash \; equiv\; \backslash \; 0\; \backslash \; (\{\backslash \; rm\; MOD\}\; \backslash \; N)$.

A stronger result is the Théorème of Wilson. N is first if and only if:

$(n-1)!\; \backslash \; \backslash \; equiv\; \backslash \; -1\; \backslash \; (\{\backslash \; rm\; MOD\}\; \backslash \; N)$

Adrien-Marie Legendre showed that the multiplicity of the prime number p in the Décomposition in product of factors first of N ! can be expressed by:

$\backslash \; sum\_\; \{i=1\}\; ^\; \{\backslash \; infty\}\; \backslash \; lfloor\; n/p^i\; \backslash \; rfloor,$

(which is defined, because the function whole Partie eliminates all the $p^i\; >\; n$).

The only factorial which is also a prime number is 2, but there exist prime numbers form $n!\; \backslash \; pm\; 1$, called factorial prime numbers.

Alternatives

Primorielle

The function Primorielle is similar to the factorial function, but into account only the product of the prime numbers takes.

Multifactorielles

In order to reduce the writing, a current notation is to use several points of exclamation to note a fonction' multifactorielle' , the product of a factor on two ( N !!), on three ( N !!!) or more.

N !! , the double factorial of N , is defined in a recurring way by:

$$

N!! = \ left \ { \begin{matrix} 1, \ qquad \ quad \ && \ mbox {if} n=0 \ mbox {or} n=1; \ \ N (N2)!! && \ mbox {if} N \ ge2. \qquad\qquad \end{matrix} \ right.

For example:

• $3!!\; =\; 3\; \backslash \; times\; 1\; =\; 3$

• $4!!\; =\; 4\; \backslash \; times\; 2\; =\; 8$
• $5!!\; =\; 5\; \backslash \; times\; 3\; \backslash \; times\; 1\; =\; 15$
• $6!!\; =\; 6\; \backslash \; times\; 4\; \backslash \; times\; 2\; =\; 48$
• $n!!\; =\; N\; \backslash \; times\; (N2)\; \backslash \; times\; (n-4)\; \backslash \; times\; \backslash \; cdots$

Certain identities rise from the definition:

$n!\; =n!!\; (n-1)!!\; \backslash ,$

$(2n)!!\; =2^nn!\; \backslash ,$

$(2n+1)!!\; =\; \{(2n+1)!\; \backslash \; over\; (2n)!!\}=\; \{(2n+1)!\; \backslash \; over2^nn!\}$

$\backslash \; Gamma\; \backslash \; left\; (n+\; \{1\; \backslash \; over2\}\; \backslash \; right)\; =\; \backslash \; sqrt\; \backslash \; pi\; \{(2n-1)!!\; \backslash over2^n\}$

Attention should be paid not to interpret N !! how factorial of N ! , which would be written ( N !)! and is a number largely larger. Certain mathematicians suggested the alternative notation N ! ₂ for the double factorial and similarly N ! _n for the other multifactorielles ones, but this use was not spread.

The double factorial is the most common alternative, but it is possible to define in a similar way triples it factorial, etc In a general way, the K E factorial, noted N ! ^{( K )} , is defined in a recurring way by:

$$

N! ^ {(K)}= \ left \ { \begin{matrix} 1, \ qquad \ qquad \ && \ mbox {if} 0 \ the n

Hyperfactorielle

The hyperfactorielle of N , noted H ( N ), is defined by:

$$

H (N) = \ prod_ {k=1} ^n k^k =1^1 \ cdot2^2 \ cdot3^3 \ cdots (n-1) ^ {n-1} \ cdot n^n.

For N = 1,2,3,4,… the values of H ( N ) are 1, 4, 108, 27.648,… ().

The hyperfactorielle function similar to the factorial function, but is produced greater numbers. Its growth is on the other hand comparable.

Superfactorielle

Neil Sloane and Simon Plouffe defined the superfactorielle in 1995 like the product of N first factorials:

$$

\ mathrm {sf} (N) = \ prod_ {k=1} ^n K! = \ prod_ {k=1} ^n k^ {n-k+1} =1^n \ cdot2^ {n-1} \ cdot3^ {N2} \ cdots (n-1) ^2 \ cdot n^1. For example, the superfactorielle one of 4 is:

$\backslash \; mathrm\; \{sf\}\; (4)\; =1!\; \backslash \; times\; 2!\; \backslash \; times\; 3!\; \backslash \; times\; 4!\; =288\; \backslash ,$

The continuation of superfactorielles begins (since N   =  0) by:

1, 1, 2, 12, 288, 34560,24883200,… to see ()

The idea was extended in 2000 by Henry Bottomley to the superduperfactorielle , produced N first superfactorielles, beginning (since N   =  0) by:

1, 1, 2, 24, 6912,238878720,5944066965504000,… to see ()

then, by recurrence, with any factorial of higher level, where the factorial of level m of N is the product of N first factorials of level m -1, i.e., by noting $f\; (N,\; m)$ factorial of N of level m :

$\backslash \; mathrm\; \{F\}\; (N,\; m)\; =\; \backslash \; mathrm\; \{F\}\; (n-1,\; m)\; \backslash \; mathrm\; \{F\}\; (N,\; M-1)$

= \ prod_ {k=1} ^n k^ {n-k+m-1 \ choose n-k}

where $\backslash \; mathrm\; \{F\}\; (N,\; 0)\; =n$ for $n>0$ and $\backslash \; mathrm\; \{F\}\; (0,\; m)\; =1$.

Superfactorielle (alternative definition)

Clifford Pickover, in its book Keys to Infinity (1995), defines the superfactorielle N , noted N $ ($ being a factorial sign! carrying an S superimposed), like:

$n\; \backslash \; \$\; \backslash \; equiv\; \backslash \; begin\; \{matrix\}\; \backslash \; underbrace\; \{N!\; ^\}\; \backslash \; \backslash \; N!\; \backslash \; end\; \{matrix\}$,

or, by using the notation of Knuth:

$n\; \backslash \; \$=\; (N!)\backslash \; uparrow\; \backslash \; uparrow\; (N!)\; \backslash ,$.

The first elements of the continuation of superfactorielles are:

$1\backslash \$=1$

$2\backslash \$=2^2=4$

$3\backslash \$=6\backslash uparrow\backslash uparrow6=6^\{6^\{6^\{6^\{6^6\}\}\}\}\; \backslash !=8.02\backslash times\; 10^\{6050\}$

Under-factorial

The function under-factorial , noted! N , is used to calculate the number of Permutation S possible as N distinct objects so that no object is in its place.

For example, there exists! N way of slipping N letters into N freed and addressed envelopes so that none the letters is in the good envelope.

There exist various ways of calculating the under-factorial

$!N\; =\; N!\; \backslash \; sum\_\; \{k=0\}\; ^n\; \backslash \; frac\; \{(-\; 1)\; ^k\}\; \{K!\}$

$!\; N\; =\; \backslash \; frac\; \{\backslash \; Gamma\; (n+1,\; -1)\}\{E\}$

Where $\backslash \; Gamma$ is the incomplete Fonction gamma and E constant mathematics.

$!N\; =\; \backslash \; left\; \backslash \; frac\; \{N!\}\{E\}\; \backslash \; right$

Where indicates the entirety nearest to X

$!N\; =!\; (n-1)\; \backslash ;\; N\; +\; (-\; 1)\; ^n$

$!\; N\; =\; (n-1)\; \backslash ;\; (!\; (n-1)\; +!\; (N2))$

$!\; N\; =\; (n-1)\; \backslash ;\; a\_\; \{N2\}$ with $\backslash ;\; a\_0\; =\; a\_1\; =\; 1$ and $a\_n\; =\; N\; \backslash ;\; a\_\; \{n-1\}\; +\; (n-1)\; \backslash ;\; a\_\; \{N2\}$

The first values of this function are:

! 1 = 0

! 2 = 1

! 3 = 2

! 4 = 9

! 5 = 44

! 6 = 265

! 7 = 1854

! 8 = 14833

! 9 = 133496

! 10 = 1334961

! 11 = 14684570

! 12 = 176214841

! 13 = 2290792932

See too

Internal bonds

• Function digamma
• Function gamma
• Prime number factorial

External bonds

• '' Fast Factorial Functions ''

Simple: Factorial

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