 Factorial

n n! 0 1 1 1 2 2 3 6 4 24 5 120 6 720 7 5040 8 40320 9 362880 10 3628800 15 1307674368000 20 2432902008176640000 25 1.5511210043×10^{25} 50 3.0414093202×10^{64} 70 1.1978571670×10^{100} 100 9.3326215444×10^{157} 170 7.2574156153×10^{306} 171 1.2410180702×10^{309} 450 1.7333687331×10^{1,000} 1000 4.0238726008×10^{2,567} 3249 6.4123376883×10^{10,000} 10000 2.8462596809×10^{35,659} 25206 1.2057034382×10^{100,000} 100000 2.8242294080×10^{456,573} 205023 2.5038989317×10^{1,000,004} 1000000 8.2639316883×10^{5,565,708} 1.0248383838×10^{98} 10^{1.0000000000×10100} 10^{100} 10^{9.9565705518×10101} 1.7976931349×10^{308} 10^{5.5336665775×10310} In mathematics, the factorial of a nonnegative 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.^{[1]}
The factorial operation is encountered in many different 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.^{[2]} The notation n! was introduced by Christian Kramp in 1808.^{[3]}
The definition of the factorial function can also be extended to noninteger arguments, while retaining its most important properties; this involves more advanced mathematics, notably techniques from mathematical analysis.
Contents
Definition
The factorial function is formally defined by
or recursively defined by
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 formulas, like 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 noninteger 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.
Applications
Although the factorial function has its roots in combinatorics, formulas involving factorials occur in many areas of mathematics.
 There are n! different ways of arranging n distinct objects into a sequence, the permutations of those objects.
 Often factorials appear in the denominator of a formula to account for the fact that ordering is to be ignored. A classical example is counting kcombinations (subsets of k elements) from a set with n elements. One can obtain such a combination by choosing a kpermutation: successively selecting and removing an element of the set, k times, for a total of

 possibilities. This however produces the kcombinations in a particular order that one wishes to ignore; since each kcombination is obtained in k! different ways, the correct number of kcombinations is
 This number is known as the binomial coefficient , because it is also the coefficient of X^{k} in (1 + X)^{n}.
 Factorials occur in algebra for various reasons, such as via the already mentioned coefficients of the binomial formula, or through averaging over permutations for symmetrization of certain operations.
 Factorials also turn up in calculus; for example they occur in the denominators of the terms of Taylor's formula, basically to compensate for the fact that the nth derivative of x^{n} is n!.
 Factorials are also used extensively in probability theory.
 Factorials can be useful to facilitate expression manipulation. For instance the number of kpermutations of n can be written as

 while this is inefficient as a means to compute that number, it may serve to prove a symmetry property of binomial coefficients:
Number theory
Factorials have many applications in number theory. In particular, n! is necessarily divisible by all prime numbers up to and including n. As a consequence, n > 5 is a composite number if and only if
A stronger result is Wilson's theorem, which states that
if and only if p is prime.
AdrienMarie Legendre found that the multiplicity of the prime p occurring in the prime factorization of n! can be expressed exactly as
This fact is based on counting the number of factors p of the integers from 1 to n. The number of multiples of p in the numbers 1 to n are given by ; however, this formula counts those numbers with two factors of p only once. Hence another factors of p must be counted too. Similarly for three, four, five factors, to infinity. The sum is finite since p^{ i} can only be less than or equal to n for finitely many values of i, and the floor function results in 0 when applied for p^{ i} > n.
The only factorial that is also a prime number is 2, but there are many primes of the form n! ± 1, called factorial primes.
All factorials greater than 0! and 1! are even, as they are all multiples of 2. Also, all factorials greater than 5! are multiples of 10 (and hence have a trailing zero as their final digit), because they are multiples of 5 and 2.
Also note that the reciprocals of factorials produce a convergent series: (see e)
Rate of growth and approximations for large n
As n grows, the factorial n! becomes larger than all polynomials and exponential functions (but slower than double exponential functions) in n.
Most approximations for n! are based on approximating its natural logarithm
The graph of the function f(n) = log n! is shown in the figure on the right. It looks approximately linear for all reasonable values of n, but this intuition is false. We get one of the simplest approximations for log n! by bounding the sum with an integral from above and below as follows:
which gives us the estimate
Hence log n! is Θ(n log n). This result plays a key role in the analysis of the computational complexity of sorting algorithms (see comparison sort).
From the bounds on log n! deduced above we get that
It is sometimes practical to use weaker but simpler estimates. Using the above formula it is easily shown that for all n we have (n / 3)^{n} < n!, and for all n ≥ 6 we have n! < (n / 2)^{n}.
For large n we get a better estimate for the number n! using Stirling's approximation:
In fact, it can be proved that for all n we have
A much better approximation for log n! was given by Srinivasa Ramanujan (Ramanujan 1988)
Computation
Computing factorials is trivial from an algorithmic point of view: successively multiplying a variable initialized to 1 by the integers 2 up to n (if any) will compute n!, provided the result fits in the variable. In functional languages, the recursive definition is often implemented directly to illustrate recursive functions.
The main difficulty in computing factorials is the size of the result. To assure that the result will fit for all legal values of even the smallest commonly used integral type (8bit signed integers) would require more than 700 bits, so no reasonable specification of a factorial function using fixedsize types can avoid questions of overflow. The values 12! and 20! are the largest factorials that can be stored in, respectively, the 32 bit and 64 bit integers commonly used in personal computers. Although floating point representation of the result allows going a bit further, it remains quite limited by possible overflow. The largest factorial that most calculators can handle is 69!, because 69! < 10^{100} < 70!. Calculators that use 3digit exponents can compute larger factorials, up to, for example, 253! ≈ 5.2×10^{499} on HP calculators and 449! ≈ 3.9×10^{997} on the TI86. The calculator seen in Mac OS X, Microsoft Excel and Google Calculator, as well as the freeware Fox Calculator, can handle factorials up to 170!, which is the largest factorial that can be represented as a 64bit IEEE 754 floatingpoint value. The scientific calculator in Windows XP is able to calculate factorials up to at least 100000!. Most software applications will compute small factorials by direct multiplication or table lookup. Larger factorial values can be approximated using Stirling's formula.
Wolfram Alpha can calculate exact results for the ceiling function and floor function applied to the binary, natural and common logarithm of n! for values of n up to 249999, and up to 20,000,000! for the Integers.
If very large exact factorials are needed, they can be computed using bignum arithmetic. In such computations speed may be gained^{[citation needed]} by not sequentially multiplying the numbers up to (or down from) n into a single accumulator, but by partitioning the sequence so that the products for each of the two parts are approximately of the same size, compute those products recursively and then multiply.
The asymptoticallybest efficiency is obtained by computing n! from its prime factorization. As documented by Peter Borwein, prime factorization allows n! to be computed in time O(n(log n log log n)^{2}), provided that a fast multiplication algorithm is used (for example, the Schönhage–Strassen algorithm).^{[4]} Peter Luschny presents source code and benchmarks for several efficient factorial algorithms, with or without the use of a prime sieve.^{[5]}
Extension of factorial to noninteger values of argument
The Gamma and Pi functions
Main article: Gamma functionBesides nonnegative integers, the factorial function can also be defined for noninteger values, but this requires more advanced tools from mathematical analysis. One function that "fills in" the values of the factorial (but with a shift of 1 in the argument) is called the Gamma function, denoted Γ(z), defined for all complex numbers z except the nonpositive integers, and given when the real part of z is positive by
Its relation to the factorials is that for any natural number n
Euler's original formula for the Gamma function was
It is worth mentioning that there is an alternative notation that was originally introduced by Gauss which is sometimes used. The Pi function, denoted Π(z) for real numbers z no less than 0, is defined by
In terms of the Gamma function it is
It truly extends the factorial in that
In addition to this, the Pi function satisfies the same recurrence as factorials do, but at every complex value z where it is defined
In fact, this is no longer a recurrence relation but a functional equation. Expressed in terms of the Gamma function this functional equation takes the form
Since the factorial is extended by the Pi function, for every complex value z where it is defined, we can write:
The values of these functions at halfinteger values is therefore determined by a single one of them; one has
from which it follows that for n ∈ N,
For example,
It also follows that for n ∈ N,
For example,
The Pi function is certainly not the only way to extend factorials to a function defined at almost all complex values, and not even the only one that is analytic wherever it is defined. Nonetheless it is usually considered the most natural way to extend the values of the factorials to a complex function. For instance, the Bohr–Mollerup theorem states that the Gamma function is the only function that takes the value 1 at 1, satisfies the functional equation Γ(n + 1) = nΓ(n), is meromorphic on the complex numbers, and is logconvex on the positive real axis. A similar statement holds for the Pi function as well, using the Π(n) = nΠ(n − 1) functional equation.
However, there exist complex functions that are probably simpler in the sense of analytic function theory and which interpolate the factorial values. For example, Hadamard's 'Gamma'function (Hadamard 1894) which, unlike the Gamma function, is an entire function.^{[6]}
Euler also developed a convergent product approximation for the noninteger factorials, which can be seen to be equivalent to the formula for the Gamma function above:
However, this formula does not provide a practical means of computing the Pi or Gamma function, as its rate of convergence is slow.
Applications of the Gamma function
The volume of an ndimensional hypersphere of radius R is
Factorial at the complex plane
Representation through the Gammafunction allows evaluation of factorial of complex argument. Equilines of amplitude and phase of factorial are shown in figure. Let . Several levels of constant modulus (amplitude) ρ = const and constant phase φ = const are shown. The grid covers range , with unit step. The scratched line shows the level .
Thin lines show intermediate levels of constant modulus and constant phase. At poles , phase and amplitude are not defined. Equilines are dense in vicinity of singularities along negative integer values of the argument.
For  z  < 1, the Taylor expansions can be used:
The first coefficients of this expansion are
n g_{n} approximation 0 1 1 1 − γ − 0.5772156649 2 0.9890559955 3 − 0.9074790760 where γ is the Euler constant and ζ is the Riemann zeta function. Computer algebra systems such as Sage (mathematics software) can generate many terms of this expansion.
Approximations of factorial
For the large values of the argument, factorial can be approximated through the integral of the digamma function, using the continued fraction representation. This approach is due to T. J. Stieltjes (1894). Writing z! = exp(P(z)) where P(z) is
Stieltjes gave a continued fraction for p(z)
The first few coefficients a_{n} are ^{[7]}
n a_{n} 0 1 / 12 1 1 / 30 2 53 / 210 3 195 / 371 4 22999 / 22737 5 29944523 / 19773142 6 109535241009 / 48264275462 There is common misconception, that or for any complex z ≠ 0. Indeed, the relation through the logarithm is valid only for specific range of values of z in vicinity of the real axis, while . The larger is the real part of the argument, the smaller should be the imaginary part. However, the inverse relation, z! = exp(P(z)), is valid for the whole complex plane apart from zero. The convergence is poor in vicinity of the negative part of the real axis. (It is difficult to have good convergence of any approximation in vicinity of the singularities). While or , the 6 coefficients above are sufficient for the evaluation of the factorial with the complex<double> precision. For higher precision more coefficients can be computed by a rational QDscheme (H. Rutishauser's QD algorithm).^{[8]}
Nonextendability to negative integers
The relation n ! = (n − 1)! × n allows one to compute the factorial for an integer given the factorial for a smaller integer. The relation can be inverted so that one can compute the factorial for an integer given the factorial for a larger integer:
Note, however, that this recursion does not permit us to compute the factorial of a negative integer; use of the formula to compute (−1)! would require a division by zero, and thus blocks us from computing a factorial value for every negative integer. (Similarly, the Gamma function is not defined for nonpositive integers, though it is defined for all other complex numbers.)
Factoriallike products and functions
There are several other integer sequences similar to the factorial that are used in mathematics:
Primorial
The primorial (sequence A002110 in OEIS) is similar to the factorial, but with the product taken only over the prime numbers.
Double factorial
A function related to the factorial is the product of all odd values up to some odd positive integer n. It is often called double factorial (even though it only involves about half the factors of the ordinary factorial, and its value is therefore closer to the square root of the factorial), and denoted by n!!.
For an odd positive integer n = 2k  1, k ≥ 1, it is
 .
For example, 9!! = 1 × 3 × 5 × 7 × 9 = 945. This notation creates a notational ambiguity with the composition of the factorial function with itself (which for n > 2 gives much larger numbers than the double factorial); this may be justified by the fact that composition arises very seldom in practice, and could be denoted by (n!)! to circumvent the ambiguity. The double factorial notation is not essential; it can be expressed in terms of the ordinary factorial by
 ,
since the denominator equals and cancels the unwanted even factors from the numerator. The introduction of the double factorial is motivated by the fact that it occurs rather frequently in combinatorial and other settings, for instance
 (2n − 1)!! is the number of permutations of 2n whose cycle type consists of n parts equal to 2; these are the involutions without fixed points.
 (2n − 1)!! is the number of perfect matchings in a complete graph K(2n).
 (2n − 5)!! is the number of unrooted binary trees with n labeled leaves.
 The value is equal to (see above)
Sometimes n!! is defined for nonnegative even numbers as well. One choice is a definition similar to the one for odd values
For example, with this definition, 8!! = 2 × 4 × 6 × 8 = 384. However, note that this definition does not match the expression above, of the double factorial in terms of the ordinary factorial, and is also inconsistent with the extension of the definition of n!! to complex numbers n that is achieved via the Gamma function as indicated below. Also, for even numbers, the double factorial notation is hardly shorter than expressing the same value using ordinary factorials. For combinatorial interpretations (the value gives, for instance, the size of the hyperoctahedral group), the latter expression can be more informative (because the factor 2^{n} is the order of the kernel of a projection to the symmetric group). Even though the formulas for the odd and even double factorials can be easily combined into
the only known interpretation for the sequence of all these numbers (sequence A006882 in OEIS) is somewhat artificial: the number of downup permutations of a set of n + 1 elements for which the entries in the even positions are increasing.
The sequence of double factorials for n = 1, 3, 5, 7, ... (sequence A001147 in OEIS) starts as
 1, 3, 15, 105, 945, 10395, 135135, ....
Some identities involving double factorials are:
Alternative extension of the double factorial
Disregarding the above definition of n!! for even values of n, the double factorial for odd integers can be extended to most real and complex numbers z by noting that when z is a positive odd integer then
The expressions obtained by taking one of the above formulas for (2n + 1)!! and (2n − 1)!! and expressing the occurring factorials in terms of the gamma function can both be seen (using the multiplication theorem) to be equivalent to the one given here.
The expression found for z!! is defined for all complex numbers except the negative even numbers. Using it as the definition, the volume of an ndimensional hypersphere of radius R can be expressed as
Multifactorials
A common related notation is to use multiple exclamation points to denote a multifactorial, the product of integers in steps of two (n!!), three (n!!!), or more. The double factorial is the most commonly used variant, but one can similarly define the triple factorial (n!!!) and so on. One can define the kth factorial, denoted by n!^{(k)}, recursively for nonnegative integers as
though see the alternative definition below.
Some mathematicians have suggested an alternative notation of n!_{2} for the double factorial and similarly n!_{k} for other multifactorials, but this has not come into general use.
With the above definition, (kn)!^{(k)} = k^{n}n!.
In the same way that n! is not defined for negative integers, and n!! is not defined for negative even integers, n!^{(k)} is not defined for negative integers evenly divisible by k.
Alternative extension of the multifactorial
Alternatively, the multifactorial z!^{(k)} can be extended to most real and complex numbers z by noting that when z is one more than a positive multiple of k then
This last expression is defined much more broadly than the original; with this definition, z!^{(k)} is defined for all complex numbers except the negative real numbers evenly divisible by k. This definition is consistent with the earlier definition only for those integers z satisfying z ≡ 1 mod k.
In addition to extending z!^{(k)} to most complex numbers z, this definition has the feature of working for all positive real values of k. Furthermore, when k = 1, this definition is mathematically equivalent to the Π(z) function, described above. Also, when k = 2, this definition is mathematically equivalent to the alternative extension of the double factorial, described above.
Quadruple factorial
The socalled quadruple factorial, however, is not the multifactorial n!^{(4)}; it is a much larger number given by (2n)!/n!, starting as
It is also equal to
Superfactorial
Main article: Large numbersNeil Sloane and Simon Plouffe defined a superfactorial in The Encyclopedia of Integer Sequences (Academic Press, 1995) to be the product of the first n factorials. So the superfactorial of 4 is
In general
Equivalently, the superfactorial is given by the formula
which is the determinant of a Vandermonde matrix.
The sequence of superfactorials starts (from n = 0) as
Alternative definition
Clifford Pickover in his 1995 book Keys to Infinity used a new notation, n$, to define the superfactorial
or as,
where the ^{(4)} notation denotes the hyper4 operator, or using Knuth's uparrow notation,
This sequence of superfactorials starts:
Here, as is usual for compound exponentiation, the grouping is understood to be from right to left:
Hyperfactorial
Occasionally the hyperfactorial of n is considered. It is written as H(n) and defined by
For n = 1, 2, 3, 4, ... the values H(n) are 1, 4, 108, 27648,... (sequence A002109 in OEIS).
The asymptotic growth rate is
where A = 1.2824... is the Glaisher–Kinkelin constant.^{[9]} H(14) = 1.8474...×10^{99} is already almost equal to a googol, and H(15) = 8.0896...×10^{116} is almost of the same magnitude as the Shannon number, the theoretical number of possible chess games. Compared to the Pickover definition of the superfactorial, the hyperfactorial grows relatively slowly.
The hyperfactorial function can be generalized to complex numbers in a similar way as the factorial function. The resulting function is called the Kfunction.
See also
 Alternating factorial
 Digamma function
 Exponential factorial
 Factorial number system
 Factorial prime
 Factorion
 Gamma function
 List of factorial and binomial topics
 Pochhammer symbol, which gives the falling or rising factorial
 Stirling's approximation
 Subfactorial
 Trailing zeros of factorial
 Triangular number, the additive analogue of factorial
Notes
 ^ Ronald L. Graham, Donald E. Knuth, Oren Patashnik (1988) Concrete Mathematics, AddisonWesley, Reading MA. ISBN 0201142368, p. 111
 ^ N. L. Biggs, The roots of combinatorics, Historia Math. 6 (1979) 109−136
 ^ Higgins, Peter (2008), Number Story: From Counting to Cryptography, New York: Copernicus, p. 12, ISBN 9781848000001 says Krempe though.
 ^ Peter Borwein. "On the Complexity of Calculating Factorials". Journal of Algorithms 6, 376–380 (1985)
 ^ Peter Luschny, FastFactorialFunctions: The Homepage of Factorial Algorithms.
 ^ Peter Luschny, Hadamard versus Euler  Who found the better Gamma function?.
 ^ Digital Library of Mathematical Functions, http://dlmf.nist.gov/5.10
 ^ Peter Luschny, On Stieltjes' Continued Fraction for the Gamma Function..
 ^ Weisstein, Eric W., "Glaisher–Kinkelin Constant" from MathWorld.
References
 Hadamard, M. J. (1894) (in French), Sur L’Expression Du Produit 1·2·3· · · · ·(n−1) Par Une Fonction Entière, OEuvres de Jacques Hadamard, Centre National de la Recherche Scientifiques, Paris, 1968, http://www.luschny.de/math/factorial/hadamard/HadamardFactorial.pdf
 Ramanujan, Srinivasa (1988), The lost notebook and other unpublished papers, Springer Berlin, p. 339, ISBN 354018726X
External links
 Approximation formulas
 All about factorial notation n!
 Weisstein, Eric W., "Factorial" from MathWorld.
 Weisstein, Eric W., "Double factorial" from MathWorld.
 Factorial at PlanetMath.
 "Double Factorial Derivations"
 Factorial calculators and algorithms
 Factorial Calculator: instantly finds factorials up to 10^14!
 Animated Factorial Calculator: shows factorials calculated as if by hand using common elementary school aglorithms
 "Factorial" by Ed Pegg, Jr. and Rob Morris, Wolfram Demonstrations Project, 2007.
 Fast Factorial Functions (with source code in Java, C#, C++, Scala and Go)
Categories: Integer sequences
 Combinatorics
 Number theory
 Gamma and related functions
 Factorial and binomial topics
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