Factorial Calculator -- n!

The factorial of a non-negative integer $n$, written $n!$, is the product of all positive integers from 1 to $n$. For example, $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$. By convention, $0! = 1$.

$0! = 1$ is defined by convention to make mathematical formulas consistent. It ensures that the binomial coefficient $C(n, 0) = 1$ (there is exactly one way to choose nothing) and that the recurrence $n! = n \times (n-1)!$ holds when $n = 1$.

Factorials grow faster than exponential functions. $10! = 3{,}628{,}800$ (7 digits), $20!$ has 19 digits, and $52!$ (ways to shuffle a deck of cards) has 68 digits. By comparison, $2^{20}$ is only about 1 million.

Stirling's approximation estimates large factorials: $n! \approx \sqrt{2\pi n} \times (n/e)^n$. The relative error is under 1% for $n \geq 10$ and under 0.1% for $n \geq 50$. It is widely used in statistics and physics where exact factorials are impractical.

Yes, using the gamma function: $\Gamma(n) = (n-1)!$ for positive integers, extended to all real numbers except non-positive integers via an integral. For example, $(1/2)! = \frac{\sqrt{\pi}}{2} \approx 0.886$, which appears in the normal distribution formula.