A factorial is an operation often encountered in mathematics used to multiply all numbers from 1 up to a given number.

Factorials are important tools in mathematics, especially for solving permutations and combinations. They are also used in probability calculations.

Factorials are denoted by an exclamation mark (!) after a number. For example, the factorial of 4 is written as 4!. This notation is used to indicate that all numbers from 1 up to and including the given number should be multiplied together. In this case, it would be

$$4! = \times 1 \times 2 \times 3 \times 4 = 24$$

The formula for the factorial of a non-negative integer n is written as n! and is defined as follows:

$$n! = n \times (n-1) \times (n-2) \times (n-3) \times \cdots \times 3 \times 2 \times 1$$

also

$$n! = n \times (n-1)!$$

Note that the **factorial of 0 is defined as 1, **because there are no positive integers less than or equal to 0.

## In summary:

0! = 1

1! = 1

2! = 2 x 1 = 2

3! = 3 x 2 x 1 = 6

4! = 4 x 3 x 2 x 1 = 24

also

4! = 4 x 3!

In conclusion, factorials are an important mathematical tool for calculating permutations, combinations and probabilities.