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:
$$n! = n \times (n-1) \times (n-2) \times (n-3) \times \cdots \times 3 \times 2 \times 1$$
$$n! = n \times (n-1)!$$
Note that the factorial of 0 is defined as 1, because there are no positive integers less than or equal to 0.
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
4! = 4 x 3!
In conclusion, factorials are an important mathematical tool for calculating permutations, combinations and probabilities.