Factorial Calculator

n! (n factorial) is the product of all positive integers from 1 to n. Example: 5! = 5 x 4 x 3 x 2 x 1 = 120. Special case: 0! = 1 by definition. Factorials grow extremely fast: 20! = 2,432,902,008,176,640,000.

0! = 1 is defined (not derived) to make combinatorics work correctly. C(n,0) = n!/(0! x n!) = 1/1 = 1, meaning there is exactly 1 way to choose 0 items from n. Without 0!=1, the combination formula would break down at boundaries.

Extremely fast: 1!=1, 5!=120, 10!=3,628,800, 15!=1.3 trillion, 20!=2.4 quintillion. At n=70, n! exceeds the number of atoms in the observable universe (~10^80). Standard 64-bit integers overflow at 21!. Python handles arbitrarily large integers.

Combinatorics: C(n,r) = n!/(r!(n-r)!), P(n,r) = n!/(n-r)!. Probability: arranging n items = n! ways. Power series in calculus: e^x = sum(x^n/n!). Statistics: binomial coefficients. Physics: Stirling approximation for large n.

For large n: ln(n!) ≈ n*ln(n) - n + 0.5*ln(2*pi*n). Or n! ≈ sqrt(2*pi*n) * (n/e)^n. This approximation is accurate to < 1% for n > 10 and gets more accurate as n grows. Useful when exact computation is infeasible.