Tool for computing factorials. Factorial n! is the product of all integer 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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Factorial of a number $ n $ 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! $.

Factorial is calculated with a multiplication

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

__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.

$$ \Gamma(n+1) = \int_0^{+\infty} t^n \exp(-t) \rm{d}t $$

and the formula that links gamma to the factorial:

$$ \forall\,n \in \mathbb{N}, \; \Gamma(n+1)=n! $$

For computing the factorial equivalent of negative numbers, 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. $$ n!\sim\sqrt{2\pi n}\left(\frac{n}{e}\right)^n $$

To calculate the product of numbers between $ n $ and $ n + m $, use factorials:

$$ \prod_{i=0}^m (n+i) = n(n+1)(n+2)\cdots(n+m) = \frac{(n+m)!}{(n-1)!} $$

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Cite as source (bibliography):

*Factorial* on dCode.fr [online website], retrieved on 2024-07-18,

- Factorial Calculator N!
- Gamma Calculator Γ(N)
- What is a factorial? (Definition)
- How to calculate a factorial?
- What is the Gamma Function?
- How to calculate a negative factorial?
- How to calculate a factorial for a decimal number?
- What is the algorithm of the factorial function?
- How to quickly compute a factorial value?
- How to calculate the product of consecutive integers?

factorial,product,exclamation,mark,gamma

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