Factorials: How to Calculate n Factorial

Calculates n! = n × (n−1) × … × 1, where n is an integer of 0 or greater. By definition 0! is 1.

Explanation

The factorial $n!$ is the product of every integer from 1 up to $n$ .

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

$n$ — an integer of 0 or greater
$n!$ — the resulting product

Factorials count arrangements. There are $n$ ways to fill the first slot in a row, $n - 1$ ways to fill the second from what is left, and so on down the line. Multiply those choices together and you get exactly $n!$ , the number of ways to order $n$ distinct objects.

Example

Take $n = 5$ .

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

Equivalently, five objects can be lined up in 120 different orders.

Notes

$0!$ is defined to be 1. There is precisely one way to arrange nothing at all, and the definition is also what keeps $n! = n \times (n-1)!$ working at $n = 1$ .
$n$ must be a whole number of 0 or greater. Negatives and fractions are rejected.
Factorials explode. $10! = 3628800$ , and $20!$ already runs to 19 digits.
This calculator stops at $n = 170$ , because $171!$ overflows the range of double-precision arithmetic.
For large $n$ the result passes the point where every digit can be stored exactly, so read the display as an approximation.