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factorial
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{| class="wikitable" style="margin:0 0 0 1em; text-align:right; float:right;"sequence {{OEIS>id=A000142}}; values specified in scientific notation are rounded to the displayed precision! n! n!
| 1
| 1
| 2
| 6
| 24
| 120
| 720
5|040}}
40|320}}
362|880}}
3800}}
39800}}
479600}}
6020|800}}
87291|200}}
1674000}}
20789000}}
355428000}}
6373728|000}}
121100832|000}}
2902176000}}
| 25
{{vale=25}}
| 50
{{vale=64}}
| 70
{{vale=100}}
| 100
{{vale=157}}
| 450
{{vale=1000}}
1|000}} {{vale=2567}}
3|249}} {{vale=10000}}
10|000}} {{vale=35659}}
25|206}} {{vale=100000}}
100|000}} {{vale=456573}}
205|023}} {{vale=1000004}}
1000}} {{vale=5565708}}
googol>{{val >|10^{10^{101.9981097754820}}
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,
5! = 5 times 4 times 3 times 2 times 1 = 120.
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. 111The 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.{{citation|last=Stedman|first=Fabian|authorlink=Fabian Stedman|title=Campanalogia|year=1677|pages=6–9|place=London}} 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):Now the nature of these methods is such, that the changes on one number comprehends [includes] the changes on all lesser numbers, ... insomuch that a compleat Peal of changes on one number seemeth to be formed by uniting of the compleat Peals on all lesser numbers into one entire body;{{sfn|Stedman|1677|p=8}}The notation n! was introduced by the French mathematician Christian Kramp in 1808.{{Citation |title=Number Story: From Counting to Cryptography |last=Higgins |first=Peter |year=2008 |publisher=Copernicus |location=New York |isbn=978-1-84800-000-1 |page=12 |pages= }} says Krempe though.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
begin{align}
n! & = prod_{k=1}^n k
& = 1 cdot 2 cdot 3 cdots (n-2) cdot (n-1) cdot n
& = n (n-1)(n-2) cdots (2)(1)
end{align}initially for integer {{math|n ≥ 1}}, and resulting in this fundamental recurrence relation:
n! = n cdot (n-1)! .
For example, one has
begin{align}5! &= 5 cdot 4!
6! &= 6 cdot 5!
50! &= 50 cdot 49! end{align}

0!

In order for this recurrence relation to be extended to n=0, it is necessary to define
0! = 1
so that
1! = 1 cdot 0! = 1 .
Other consequences that indicate defining 0! = 1 and the convention that the product of no numbers at all is 1 are:
  • There is exactly one permutation of zero objects (with nothing to permute, "everything" is left in place).
  • It makes many identities in combinatorics valid for all applicable sizes. The number of ways to choose 0 elements from the empty set is


tbinom{0}{0} = tfrac{0!}{0!0!} = 1 .
More generally, the number of ways to choose (all) n elements among a set of n is
tbinom{n}{n} = tfrac{n!}{n!0!} = 1 .
  • It allows for the compact expression of many formulae, such as the exponential function, as a power series:


e^x = sum_{n = 0}^infty frac{x^n}{n!}.

Factorial of a non-integer

The factorial function can also be defined for non-integer values using more advanced mathematics (the gamma function n! = Gamma(n+1) ), 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.BOOK, Beyond Infinity: An expedition to the outer limits of the mathematical universe, Cheng, Eugenia, 2017-03-09, Profile Books, 9781782830818, en, Eugenia Cheng, BOOK, The Book of Numbers, Conway, John H., Guy, Richard, 1998-03-16, Springer Science & Business Media, 9780387979939, en, John Horton Conway, Richard K. Guy,
  • 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 k-combinations (subsets of k elements) from a set with n elements. One can obtain such a combination by choosing a k-permutation: successively selecting and removing an element of the set, k times, for a total of


n^{underline k}=n(n-1)(n-2)cdots(n-k+1)
possibilities. This however produces the k-combinations in a particular order that one wishes to ignore; since each k-combination is obtained in k! different ways, the correct number of k-combinations is
frac{n^{underline k}}{k!}=frac{n(n-1)(n-2)cdots(n-k+1)}{k(k-1)(k-2)cdots(1)}.
This number is knownBOOK, The Art of Computer Programming: Volume 1: Fundamental Algorithms, Knuth, Donald E., 1997-07-04, Addison-Wesley Professional, 9780321635747, en, Donald Knuth, as the binomial coefficient tbinom nk, because it is also the coefficient of Xk in {{nowrap|(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,WEB,weblink 18.01 Single Variable Calculus, Lecture 37: Taylor Series, Fall 2006, MIT OpenCourseWare,weblink" title="web.archive.org/web/20121706133400weblink">weblink 2018-04-26, no, 2017-05-03, where they are used as compensation terms due to the n-th derivative of xn being equivalent to n!.
  • Factorials are also used extensively in probability theory.BOOK, Statistical Physics of Particles, Kardar, Mehran, 2007-06-25, Cambridge University Press, 9780521873420, English, Chapter 2: Probability, Mehran Kardar,
  • Factorials can be useful to facilitate expression manipulation. For instance the number of k-permutations of n can be written as


n^{underline k}=frac{n!}{(n-k)!};
while this is inefficient as a means to compute that number, it may serve to prove a symmetry property of binomial coefficients:
binom nk=frac{n^{underline k}}{k!}=frac{n!}{(n-k)!k!} = frac{n^{underline{n-k}}}{(n-k)!} = binom n{n-k}.
  • The factorial function can be shown, using the power rule, to be


begin{align}
n! &= D^n(x^n)
&= frac{d^n}{dx^n}(x^n)
end{align}
where D^nx^n is the Euler's notation for the nth derivative of x^n .WEB,weblink 18.01 Single Variable Calculus, Lecture 4: Chain rule, higher derivatives, Fall 2006, MIT OpenCourseWare,weblink" title="web.archive.org/web/20121706133400weblink">weblink 2018-04-26, no, 2017-05-03,

Rate of growth and approximations for large n

(File:Log-factorial.svg|upright=1.35|thumb|right|Plot of the natural logarithm of the factorial)As n grows, the factorial n! increases faster 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
ln n! = sum_{x=1}^n ln x.
The graph of the function f(n) = ln 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 ln n! by bounding the sum with an integral from above and below as follows:
int_1^n ln x , dx leq sum_{x=1}^n ln x leq int_0^n ln (x+1) , dx
which gives us the estimate
nlnleft(frac{n}{e}right)+1 leq ln n! leq (n+1)lnleft( frac{n+1}{e} right) + 1.
Hence ln n! sim nln n (see Big O notation). This result plays a key role in the analysis of the computational complexity of sorting algorithms (see comparison sort). From the bounds on ln n! deduced above we get that
left(frac neright)^n e leq n! leq left(frac{n+1}eright)^{n+1} e.
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.(File:Mplwp factorial gamma stirling.svg|thumb|right|upright=1.35|Comparison of Stirling's approximation with the factorial)For large n we get a better estimate for the number n! using Stirling's approximation:
n!simsqrt{2pi n}left(frac{n}{e}right)^n.
This in fact comes from an asymptotic series for the logarithm, and n factorial lies between this and the next approximation:
sqrt{2pi n}left(frac{n}{e}right)^n


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