Factorial Calculator

Factorial is one of the most fundamental operations in combinatorics, probability, and discrete mathematics. It represents the number of ways to arrange a set of distinct objects.

Factorial definition: n! = n × (n−1) × (n−2) × ... × 2 × 1 0! = 1 (by convention, since there is exactly one way to arrange zero objects)

Worked examples:

• 5! = 5 × 4 × 3 × 2 × 1 = 120
• 10! = 3,628,800
• 20! = 2,432,902,008,176,640,000

Permutations, the ordered arrangements: P(n, r) = n! / (n − r)!

How many ways can you arrange 3 books chosen from a shelf of 7? P(7, 3) = 7! / 4! = 5,040 / 24 = 210 arrangements

Combinations, the unordered selections: C(n, r) = n! / (r! × (n − r)!)

How many ways can you choose 3 students from a class of 20? C(20, 3) = 20! / (3! × 17!) = 6,840 / 6 = 1,140 combinations

Stirling’s approximation (for very large n): n! ≈ √(2πn) × (n/e)^n

It undershoots by roughly 1/(12n), so at n = 100 it is about 0.08% low and the leading digits are already right. The calculator shows it beside the exact answer so you can watch the gap close.

About precision. This page works in arbitrary-precision integers, so every digit it prints is exact, all the way to 1000!. That matters more than it sounds. An ordinary JavaScript number stops being able to hold a factorial exactly somewhere around 22!, and after that you get a value that looks right and is quietly wrong in its lower digits. If a calculator hands you 25! as 15,511,210,043,330,985,984,000,000 it is telling the truth; if it hands you 1.5511210043331e+25 it is telling you it gave up.

Real-world applications:

• Card games: a shuffled deck has 52! ≈ 8.07 × 10⁶⁷ orderings, which is why any well-shuffled deck has almost certainly never existed before
• Probability: exact odds in lotteries, where C(49, 6) = 13,983,816 is the classic
• Statistics: binomial coefficients, the C(n, r) tab on this page
• Computer science: algorithm complexity analysis, e.g. O(n!) brute-force over permutations
• Scheduling: number of possible task orderings

Growth rate: Factorial grows faster than exponential. 20! has 19 digits; 100! has 158 digits; 1000! has 2,568.