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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