Tool for computing factorials. Factorial n! is the product of all integers numbers (not zero) inferior or equal to n, it is symbolized by an exclamation point juxtaposed after the number.

Factorial - dCode

Tag(s) : Arithmetics

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Tool for computing factorials. Factorial n! is the product of all integers numbers (not zero) inferior or equal to n, it is symbolized by an exclamation point juxtaposed after the number.

**Factorial** of a number $ n $ is calculated with a multiplication: it is the product of the positive integers numbers (not null) less or equal to $ n $.

The usual notation to indicate a **factorial** is the exclamation mark positioned after the number. The **factorial** of $ n $ is noted $ n! $.

$$ n!=\prod_{k=1}^n k = 1 \times 2 \times 3 \times \cdots \times n $$

__Example:__ $$ 4! = 1 \times 2 \times 3 \times 4 = 24 $$

__Example:__ The number of ways to sort a set of 52 cards is worth $ 52! = 1 \times 2 \times \dots \times 51 \times 52 = \\ 806581751709438785716606368564037\\66975289505440883277824000000000000 \\ \approx 8.0658 \times 10^{67} $$

Note that the **factorial** of zero is equal to one: $ 0! = 1 $

__Example:__ Here are the values of the first **factorials** $$ 0! = 1 \\ 1! = 1 \\ 2! = 2 \\ 3! = 6 \\ 4! = 24 \\ 5! = 120 \\ 6! = 720 \\ 7! = 5040 \\ 8! = 40320 \\ 9! = 362880 \\ 10! = 3628800 $$

Euler-Gamma is an extension of the **factorial** function over the complex numbers set. dCode offers calculation over the Reals. $$ \forall\,n \in \mathbb{N}, \; \Gamma(n+1)=n! $$

For computing the **factorial** equivalent of negative number, use the Gamma function.

Pour computing the **factorial** equivalent of fraction or decimal numbers, use the Gamma function.

The **factorial** algorithm with a loop:`function fact(n) {`

f = 1

if (n >= 2) {

for (i = 2 ; i < n; i++) {

f = f * i

}

}

return f

}

The recursive **factorial** algorithm: `function fact(n) {`

if (n <= 1)

return 1

else

return fact(n-1)*n

}

For large numbers, it is possible to estimate the value of $ n! $ with a good precision using the Stirling formula. $$ n!\sim\sqrt{2\pi n}\left(\frac{n}{e}\right)^n $$

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