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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

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: $$145 = 1! + 4! + 5! = 5 × 29$$ and $$40,585 = 4! + 0! + 5! + 8! + 5! = 5 × 8,117$$.

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

• 153 is the sum of the factorials of the first five positive numbers, and also the exponent in the short scale quinquagintillion.
• The first carrier frequency in the longwave radio band is at 153 kHz.
• It is also the number of fish in the second miraculous catch of fish.
• 154 is the sum of the factorials of the first six nonnegative numbers, and 154! + 1 is a factorial prime.
• 720 is an integer equal to 6!, the factorial of 6. Consequently, it is the order of the symmetric group of degree 6, which is isomorphic to B2(2), and has an outer automorphism.
• It is also the number of degrees in a hexagon.
• Furthermore, it is the number of hours in a 30-day month (April, June, September or November) not containing a DST transition.
• And some high-definition television services have 720 visible scan lines.
• Finally, it is also the number of pixels in a standard-definition television scan line.
• 5,040 is an integer equal to 7!. It is the largest known factorial number which is the predecessor of a square number: 7! = 5,040 = 5,041−1 = 712−1.
• Plato mentioned in his Laws that 5,040 is a convenient number to use for dividing many things (including both the citizens and the land of a city-state or polis) into lesser parts, making it an ideal number for the number of citizens (heads of families) making up a polis. He remarks that this number can be divided by all the (natural) numbers from 1 to 12 with the single exception of 11 (however, it is not the smallest number to have this property; 2,520 is).
• 479,001,600 is equal to $$12!$$, and therefore the number of possible tone rows in the twelve-tone technique.
• 1,124,000,727,777,607,680,000 is a positive integer equal to $$22!$$. It is notable in computer science for being the largest factorial number which can be represented exactly in the double floating-point format (which has a 53-bit significand).
• 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.
• One billion factorial is approximately 1.57637137 × 108,565,705,531.

Variation

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 $$x!0 = x$$.[5]

$$x!n$$ 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

// 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