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In mathematics, the factorial of a positive integer

$ n $ , denoted by

$ n! $ , is the product of all positive integers less than or equal to Template:Mvar. For example,

$ 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 500 \,. $

The value of 0! is 1, according to the convention for an empty product.

The factorial operation is encountered in many areas of mathematics, notably in combinatorics, algebra, and mathematical analysis. Its most basic use counts the possible distinct sequences -- the permutations -- of

$ n $ distinct objects: there are

$ n! $ .

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.

Table of valuesEdit

These are selected members of the factorial sequence, the values > 20! are written in scientific notation and rounded to 10 significant digits.

$ n $$ n! $
01
11
22
36
420
5X0
6500
72 E00
81E 400
9156 000
X1 270 000
E11 450 000
10114 500 000
111 259 500 000
1214 X8E X00 000
13191 529 600 000
142 41X E88 000 000
1533 X86 734 000 000
164EX 09X E00 000 000
177 X8E 383 500 000 000
18111 XXX 198 400 000 000
191 E04 0E9 1E7 000 000 000
1X36 275 969 72X 000 000 000
1E68E 041 404 X52 000 000 000
2011 5X0 828 098 X40 000 000 000
303.789 X09 464 12 × 1032
404.69X 128 83E 73 × 1048
509.702 57X 114 20 × 1063
601.649 42E 94E XX × 1080
701.978 X87 570 6E × 1099
80E.894 332 8E8 52 × 10E6
833.845 27E 248 23 × 10100
902.447 652 178 27 × 10115
X01.977 E26 887 95 × 10134
E04.672 692 560 80 × 10153
1002.X10 EX1 516 50 × 10173
200X.581 X35 0XE 81 × 10391
3006.258 191 777 10 × 10616
4001.739 E34 723 E5 × 10877
5001.EE3 226 EXE 9E × 10E2X
536E.370 463 766 4E × 101 001
6001.090 525 435 91 × 101 1E1
7003.259 2X0 904 8X × 101 481
8001.864 X09 8E9 1X × 101 75X
9001.8X7 123 575 22 × 101 X42
X005.4E0 E88 62E X5 × 102 130
E00X.373 X3X 427 34 × 102 424
1 0003.288 37X XX0 39 × 102 722
3 9X92.303 X87 E81 15 × 1010 001
10 0004.3E1 613 493 EE × 1037 209
2E 8183.272 320 689 04 × 10100 002
100 0003.62E 367 978 48 × 10472 075
250 X021.4E8 264 920 30 × 101 000 000
1 000 0001.037 571 E51 99 × 105 720 72E

DefinitionEdit

The factorial function is defined by the product

$ n! = 1 \cdot 2 \cdot 3 \cdots (n-2) \cdot (n-1) \cdot n, $

for integer Template:Math. This may be written in the Pi product notation as

$ n! = \prod_{i = 1}^n i. $

From these formulas, one may derive the recurrence relation

$ n! = n \cdot (n-1)! . $

For example, one has

$ \begin{align} 4! &= 4 \cdot 3! \\ 6! &= 6 \cdot 5! \\ 100! &= 100 \cdot \mathcal{E}\mathcal{E}! \end{align} $

and so on.

Factorial of zeroEdit

The factorial of 0,

$ 0! $ , is 1.

There are several motivations for this definition:

  • For Template:Math, the definition of Template:Math as a product involves the product of no numbers at all, and so is an example of the broader convention that the product of no factors is equal to the multiplicative identity (see empty product).
  • There is exactly one permutation of zero objects (with nothing to permute, the only rearrangement is to do nothing).
  • It makes many identities in combinatorics valid for all applicable sizes. The number of ways to choose 0 elements from the empty set is given by the binomial coefficient
$ \binom{0}{0} = \frac{0!}{0!0!} = 1 $.
More generally, the number of ways to choose all Template:Mvar elements among a set of Template:Mvar is
$ \binom{n}{n} = \frac{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!}. $
  • It extends the recurrence relation to 0.

Factorial of a non-integerEdit

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

ApplicationsEdit

Although the factorial function has its roots in combinatorics, formulas involving factorials occur in many areas of mathematics.

$ (n-0)(n-1)(n-2)\cdots\left(n-(k-1)\right) = \tfrac{n!}{(n-k)!} = n^{\underline k} $
possibilities. This however produces the Template:Mvar-combinations in a particular order that one wishes to ignore; since each Template:Mvar-combination is obtained in Template:Math different ways, the correct number of Template:Mvar-combinations is
$ \frac{n(n-1)(n-2)\cdots(n-k+1)}{k(k-1)(k-2)\cdots 1} = \frac{n^{\underline k}}{k!}= \frac{n!}{(n-k)!k!} = \binom {n}{k}. $
This number is known[3] as the binomial coefficient, because it is also the coefficient of Template:Math in Template:Math. The term$ n^{\underline k} $ is often called a falling factorial (pronounced "n to the falling k").
$ 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[2][3] 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
$ n! = D^n\,x^n = \frac{d^n}{dx^n}\,x^n $
where Template:Math is Euler's notation for the Template:Mvarth derivative of Template:Math.[6]

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