Factorial Calculator – Compute n! Instantly Online
Calculate the factorial of any non-negative integer instantly with our free online factorial calculator. Supports large factorials with exact results.
Understanding Factorials
The factorial of a number n (written as n!) is the product of all positive integers from 1 up to n. It's one of the most fundamental operations in combinatorics, probability, and calculus.
The Factorial Formula
n! = n × (n-1) × (n-2) × ... × 2 × 1For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. By definition, 0! = 1 — this isn't a mistake, it's a convention that makes many formulas work cleanly.
Worked Examples
Example 1: 5!
5! = 5 × 4 × 3 × 2 × 1 = 120Small factorials like this show up in permutation problems — like arranging 5 books on a shelf.
Example 2: 10!
10! = 3,628,800Factorials grow fast. 10! is already over 3.6 million. This is why they're useful for counting large numbers of arrangements.
Example 3: 20!
20! = 2,432,902,008,176,640,00020! is about 2.4 quintillion — more than the number of grains of sand on Earth. This is why we use BigInt for exact calculations.
A Quick Fact
The factorial notation was introduced by French mathematician Christian Kramp in 1808. Before that, mathematicians wrote out "factorial n" in words. The exclamation point is surprisingly fitting — factorials do tend to produce astonishingly large numbers.
Common Questions
Why is 0! equal to 1?
It's not arbitrary — defining 0! = 1 makes formulas work. For example, the number of ways to arrange 0 objects is 1 (do nothing). The binomial coefficient C(n,0) = n!/(0!×n!) only works if 0! = 1.
How fast do factorials grow?
Faster than exponential. 10! is 3.6 million, 20! is 2.4 quintillion, and 70! is about 1.2 × 10^100 — larger than the number of atoms in the observable universe.
Can I calculate factorials of negative numbers?
Not with the standard definition. The factorial is only defined for non-negative integers. There's a related function called the gamma function that extends the idea to negative and non-integer values, but that's beyond this calculator's scope.
What's the largest factorial I can calculate here?
This calculator handles up to 500!. The result has 1,135 digits. JavaScript's BigInt handles it exactly, but displaying larger values becomes impractical.
Where do factorials show up in real problems?
Anywhere you count arrangements or permutations. Password combinations, seating arrangements, card shuffles, scheduling problems — if you're counting "how many ways can I order these things," factorials are involved.
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