## FANDOM

10,098 Pages

Factorial
TypeCombinatorial
Based onMultiplication
Growth rate$$f_{2}(n)$$

The factorial is a function applied to whole numbers, defined as[1][2]

$$n! = \prod^n_{i = 1} i = n \cdot (n - 1) \cdot \ldots \cdot 4 \cdot 3 \cdot 2 \cdot 1.$$

For example, $$6! = 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 720$$. It is equal to the number of ways $$n$$ distinct objects can be arranged, because there are $$n$$ ways to place the first object, $$n - 1$$ ways to place the second object, and so forth. The special case $$0! = 1$$ has been set by definition; there is one way to arrange zero objects.

Before the notation $$n!$$ was invented, $$n$$ was common.

The function can be defined recursively as $$0! = 1$$ and $$n! = n \cdot (n - 1)!$$. The first few values of $$n!$$ for $$n = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11$$ are 1, 1 , 2, 6, 24, 120, 720, 5,040, 40,320, 362,880, 3,628,800, and 39,916,800.

## Properties Edit

The sum of the reciprocals of the factorials is $$\sum^{\infty}_{i = 0} \frac{1}{i!} = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \cdots = 2.71828182845904\ldots$$, a mathematical constant better known as $$e$$. In fact, $$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$$, which illustrates the important property that $$\frac{d}{dx}e^x = e^x$$.

Because $$n! = \Gamma (n + 1)$$ (where $$\Gamma (x)$$ is the gamma function), $$n! = \int^{\infty}_0 e^{-t} \cdot t^{n} dt$$. This identity gives us factorials of positive real numbers (and negative non-integer real numbers), not limited to integers:

• $$\left(\frac{1}{2}\right)! = \frac{\sqrt{\pi}}{2}$$
• $$\left(-\frac{1}{2}\right)! = \sqrt{\pi}$$

The most well-known approximation of n! is $$n!\approx \sqrt{2\pi n}(\frac{n}{e})^n$$, and it's called Stirling's approximation.

In base 10, only two non-trivial numbers are equal to the sum of the factorials of their digits: $$145 = 1! + 4! + 5!$$ and $$40,585 = 4! + 0! + 5! + 8! + 5!$$.

The number of zeroes at the end of the decimal expansion of $$n!$$ is $$\sum_{k = 1} \lfloor n / 5^k\rfloor$$.[3] For example, 10,000! has 2,000 + 400 + 80 + 16 + 3 = 2,499 zeroes.

## Specific numbers Edit

• 70! is the smallest factorial which is greater than googol, while 69! still has only 99 digits.
• One hundred factorial's decimal expansion is shown below.
93,326,215,443,944,152,681,699,238,856,266,700,490,715,968,264,381,621,468,592,963,895,217,599,993,229,915,608,941,463,976,156,518,286,253,697,920,827,223,758,251,185,210,916,864,000,000,000,000,000,000,000,000
• In scientific notation, this is approximately 9.3326215443 × 10157. It seems to be approximately googol3/2, although it is almost 100 million times larger.
• Lawrence Hollom calls 200! faxul.
• One thousand factorial is about 4.0238726007 × 102,567.
• Aarex Tiaokhiao has proposed the name Myriadbang for 10,000!.
• One million factorial is approximately 8.2639317 × 105,565,708.

## Variation Edit

Aalbert Torsius defines a variation on the factorial, where $$x!n = \prod^{x}_{i = 1} i!(n - 1) = 1!(n - 1) \cdot 2!(n - 1) \cdot \ldots \cdot x!(n - 1)$$ and $$x!0 = x$$.[4]

$$x!n$$ is pronounced "nth level factorial of x." $$x!1$$ is simply the ordinary factorial and $$x!2$$ is Sloane and Plouffe's superfactorial $$x\$$.

The special case $$x!x$$ is a function known as the Torian.

## Pseudocode Edit

// Standard factorial function
function factorial(z):
result := 1
for i from 1 to z:
result := result * i
return result

// Generalized factorial, using Lanczos approximation for gamma function

g := 7
coeffs := [0.99999999999980993, 676.5203681218851, -1259.1392167224028, 771.32342877765313, -176.61502916214059, 12.507343278686905, -0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7]

function factorialReal(z):
ag := coeffs[0]
for i from 1 to g + 1:
ag := ag + coeffs[i] / (z + i)
zg := z + g + 0.5
return sqrt(2 * pi) * zgz + 0.5 * e-zg * ag

// Torsius' factorial extension
function factorialTorsius(z, x):
if x = 0:
return z
if x = 1:
return factorial(z)
result := 1
for i from 1 to z:
result := result * factorialTorsius(i, x - 1)
return result


## Sources Edit

Factorials
Main article: Factorial
Multifactorials: Double factorial · Multifactorial
Falling and rising: Falling factorial · Rising factorial
Other mathematical variants: Alternating factorial · Hyperfactorial · q-factorial · Roman factorial · Subfactorial · Weak factorial · Bouncing Factorial
Tetrational growth: Exponential factorial · Expostfacto function · Superfactorial by Clifford Pickover
Array-based extensions: Hyperfactorial array notation · Nested factorial notation
Other googological variants: · Superfactorial by Sloane and Plouffe · Torian · Factorexation · Mixed factorial · Bouncing Factorial

Large numbers in combinatorics