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! $.

$$ n!=\prod_{k=1}^n k $$

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 = \\ 80658175170943878571660636856403766975289505440883277824000000000000 $$

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

Use the Gamma function.

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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