In mathematics, the factorial of a positive integer

, denoted by

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

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

distinct objects: there are

.

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 values

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

 0 1 1 1 2 2 3 6 4 20 5 X0 6 500 7 2 E00 8 1E 400 9 156 000 X 1 270 000 E 11 450 000 10 114 500 000 11 1 259 500 000 12 14 X8E X00 000 13 191 529 600 000 14 2 41X E88 000 000 15 33 X86 734 000 000 16 4EX 09X E00 000 000 17 7 X8E 383 500 000 000 18 111 XXX 198 400 000 000 19 1 E04 0E9 1E7 000 000 000 1X 36 275 969 72X 000 000 000 1E 68E 041 404 X52 000 000 000 20 11 5X0 828 098 X40 000 000 000 30 3.789 X09 464 12 × 1032 40 4.69X 128 83E 73 × 1048 50 9.702 57X 114 20 × 1063 60 1.649 42E 94E XX × 1080 70 1.978 X87 570 6E × 1099 80 E.894 332 8E8 52 × 10E6 83 3.845 27E 248 23 × 10100 90 2.447 652 178 27 × 10115 X0 1.977 E26 887 95 × 10134 E0 4.672 692 560 80 × 10153 100 2.X10 EX1 516 50 × 10173 200 X.581 X35 0XE 81 × 10391 300 6.258 191 777 10 × 10616 400 1.739 E34 723 E5 × 10877 500 1.EE3 226 EXE 9E × 10E2X 536 E.370 463 766 4E × 101 001 600 1.090 525 435 91 × 101 1E1 700 3.259 2X0 904 8X × 101 481 800 1.864 X09 8E9 1X × 101 75X 900 1.8X7 123 575 22 × 101 X42 X00 5.4E0 E88 62E X5 × 102 130 E00 X.373 X3X 427 34 × 102 424 1 000 3.288 37X XX0 39 × 102 722 3 9X9 2.303 X87 E81 15 × 1010 001 10 000 4.3E1 613 493 EE × 1037 209 2E 818 3.272 320 689 04 × 10100 002 100 000 3.62E 367 978 48 × 10472 075 250 X02 1.4E8 264 920 30 × 101 000 000 1 000 000 1.037 571 E51 99 × 105 720 72E

## Definition

The factorial function is defined by the product

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

From these formulas, one may derive the recurrence relation

For example, one has

and so on.

### Factorial of zero

The factorial of 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
.
More generally, the number of ways to choose all Template:Mvar elements among a set of Template:Mvar is
.
• It allows for the compact expression of many formulae, such as the exponential function, as a power series:
• It extends the recurrence relation to 0.

### Factorial of a non-integer

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.

## Applications

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

• There are Template:Math different ways of arranging Template:Mvar distinct objects into a sequence, the permutations of those objects.[1][2]
• 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 Template:Mvar-combinations (subsets of Template:Mvar elements) from a set with Template:Mvar elements. One can obtain such a combination by choosing a Template:Mvar-permutation: successively selecting and removing one element of the set, Template:Mvar times, for a total of
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
This number is known[3] as the binomial coefficient, because it is also the coefficient of Template:Math in Template:Math. The term is often called a falling factorial (pronounced "n to the falling k").
• 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,[4] where they are used as compensation terms due to the Template:Mvarth derivative of Template:Math being equivalent to Template:Math.
• Factorials are also used extensively in probability theory[5] and number theory (see below).
• Factorials can be useful to facilitate expression manipulation. For instance the number of Template:Mvar-permutations of Template:Mvar can be written as
while this is inefficient as a means to compute that number, it may serve to prove a symmetry property[2][3] of binomial coefficients:
• The factorial function can be shown, using the power rule, to be
where Template:Math is Euler's notation for the Template:Mvarth derivative of Template:Math.[6]
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