# Factorial

## Definition

The product of a non-negative integer and all the integers below it is called the factorial.

The term factorial is related to factors which are multiplying elements. So, the factorial of a number represents the product of the number and all the numbers below to that number but all the numbers are positive integers except zero.

### Representation

Exclamation mark (!) is used to denote the factorial operation in mathematics. It is written after the integer.

##### Example

For example, let us find the value of $5!$.

$5! = 5 \times 4 \times 3 \times 2 \times 1$

The factorial symbol represents the product of all the numbers which are below a particular number.

$\implies 5! = 120$

Observe the following examples for better understanding.

$1! = 1$

$2! = 2 \times 1 = 2$

$3! = 3 \times 2 \times 1 = 6$

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

$5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$

$6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720$

$7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040$

You can calculate factorial of any number with the multiplication process by decreasing the number by $1$ until you reach integer $1$.

Remember, the value of $0!$ is also equal to $1$.

#### General form

$n! = n.(n-1).(n-2) \cdots 3.2.1$
(or)
$n! = 1.2.3 \cdots (n-2).(n-1).n$

The factorial of a number can also be expressed in product of sequence form algebraically.

$$n! = \prod_{\displaystyle x = 1}^{\displaystyle n} x$$

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Jun 26, 2023

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