Factorial Calculator
Calculate n! (n factorial) for any non-negative integer. See the full expansion, digit count, and applications in permutations and combinations.
โ Factorial Calculator
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๐ก 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 ร 4 ร 3 ร 2 ร 1 = 120. By convention, 0! = 1. Factorials grow extraordinarily fast โ 10! = 3,628,800 and 20! = 2,432,902,008,176,640,000 (over 2.4 quintillion).
Factorials are the foundation of combinatorics โ the mathematics of counting. Permutations (ordered arrangements) use the formula P(n,r) = n!/(nโr)!, and combinations (unordered selections) use C(n,r) = n!/(r!(nโr)!). These formulas appear in probability theory, statistics, computer science algorithms, and even card games and lottery calculations.
Stirling's approximation (n! โ โ(2ฯn) ร (n/e)โฟ) is used to estimate factorials of very large numbers. The gamma function ฮ(n) = (nโ1)! extends factorials to non-integer and even complex numbers, which is essential in advanced mathematics, physics, and engineering.
Where:
- n! = Product of all positive integers from 1 to n
- 0! = Defined as 1 (by convention)
- P(n,r) = Permutations = n!/(nโr)!
- C(n,r) = Combinations = n!/(r!(nโr)!)
๐ Worked Example
6!
6 ร 5 ร 4 ร 3 ร 2 ร 1= 720
C(10, 3) โ choose 3 from 10
10! / (3! ร 7!) = 720/(6ร5040)= 120 ways