## Factorial Calculator N!

## Gamma Calculator Γ(N)

## Answers to Questions

### How to calculate a factorial?

**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** $$ 0! = 1 \\ 1! = 1 \\ 2! = 2 \\ 3! = 6 \\ 4! = 24 \\ 5! = 120 \\ 6! = 720 \\ 7! = 5040 \\ 8! = 40320 \\ 9! = 362880 \\ 10! = 3628800 $$

### What is the Gamma Function?

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

### How to calculate a negative factorial?

Use the Gamma function.

### How to quickly compute a factorial value?

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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