Factorial Calculator

Calculate n! (n factorial) for any non-negative integer. See the full expansion, digit count, and applications in permutations and combinations.

❗ Factorial Calculator

Result

10!3,628,800

Digits7

Expansion10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1

Permutations P(10,3)720

Combinations C(10,3)120

💡 What is a Factorial and What is it Used For?

The factorial of a non-negative integer n, written as n!, is the product of all positive integers from 1 to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. By convention, 0! = 1. Factorials grow extraordinarily fast — 10! = 3,628,800 and 20! = 2,432,902,008,176,640,000 (over 2.4 quintillion).

Factorials are the foundation of combinatorics — the mathematics of counting. Permutations (ordered arrangements) use the formula P(n,r) = n!/(n−r)!, and combinations (unordered selections) use C(n,r) = n!/(r!(n−r)!). These formulas appear in probability theory, statistics, computer science algorithms, and even card games and lottery calculations.

Stirling's approximation (n! ≈ √(2πn) × (n/e)ⁿ) is used to estimate factorials of very large numbers. The gamma function Γ(n) = (n−1)! extends factorials to non-integer and even complex numbers, which is essential in advanced mathematics, physics, and engineering.

10! = 3,628,800. This means there are 3,628,800 different ways to arrange 10 distinct objects in a line. And 52! (shuffled deck of cards) has 80 digits!

n! = n × (n−1) × (n−2) × ... × 2 × 1

Where:

n! = Product of all positive integers from 1 to n
0! = Defined as 1 (by convention)
P(n,r) = Permutations = n!/(n−r)!
C(n,r) = Combinations = n!/(r!(n−r)!)

📝 Worked Example

6 × 5 × 4 × 3 × 2 × 1

= 720

C(10, 3) — choose 3 from 10

10! / (3! × 7!) = 720/(6×5040)

= 120 ways