Factorial Calculator

What is a Factorial?

The factorial of a non-negative integer n, written n!, is the product of all positive integers from 1 up to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. By definition, 0! = 1. Factorials grow extremely quickly, so this calculator uses exact big-integer arithmetic to give you the precise value even for large n.

Where Factorials Are Used

• Permutations: the number of ways to arrange n distinct items in order is n!.
• Combinations: the binomial coefficient C(n, k) = n! / (k! × (n − k)!) counts how many ways you can choose k items from n.
• Probability & statistics: factorials appear in distributions such as the Poisson and binomial.
• Series & calculus: Taylor and Maclaurin series use factorials in their denominators.

Why Does 0! = 1?

Defining 0! = 1 keeps important formulas consistent. There is exactly one way to arrange zero objects (the empty arrangement), so the count should be 1. It also makes the recursive rule n! = n × (n − 1)! work for n = 1, since 1! = 1 × 0! = 1, and it keeps the combination formula C(n, 0) = 1 valid.

How Fast Do Factorials Grow?

Factorials grow faster than any exponential function. 10! is already 3,628,800, and 20! exceeds 2.4 quintillion. By the time you reach 70!, the result has about 100 digits. For very large inputs this calculator reports the digit count and a scientific-notation approximation alongside the exact value.

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