The factorial of a number — multiplying it by every positive whole number below it — grows explosively fast, and underlies how mathematicians count arrangements and possibilities. This tool calculates factorials instantly, even for numbers where the result would take you ages by hand.
A notation invented specifically to save space
The factorial notation (the exclamation mark, as in 5!) was introduced by French mathematician Christian Kramp in 1808, explicitly as a shorthand convenience for a calculation — multiplying a number by every positive integer below it down to 1 — that mathematicians had already been using in combinatorics and probability calculations for some time before that specific notation existed. The dramatic, explosive growth rate of factorials (10! already exceeds 3.6 million, and 20! exceeds 2.4 quintillion) is precisely why the notation's compactness mattered: writing out "20 × 19 × 18 × ... × 2 × 1" repeatedly in mathematical texts would have been genuinely cumbersome compared to simply writing "20!"
How factorial calculation works
The tool multiplies your input number by every positive whole number smaller than it, all the way down to 1 — 5! = 5 × 4 × 3 × 2 × 1 = 120 — a calculation whose result grows so rapidly that even relatively modest starting numbers (like 20 or 25) produce results in the billions or quintillions, a phenomenon mathematicians call "factorial growth," among the fastest-growing common functions in mathematics.
Where factorial calculations are genuinely useful
- Combinatorics and probability — factorials are foundational to calculating permutations (how many ways a set of items can be arranged) and combinations (how many ways items can be selected), core concepts across statistics, probability and combinatorics coursework.
- Computer science and algorithm complexity — factorial growth appears as a benchmark "worst case" complexity class in algorithm analysis, representing problems that become computationally infeasible remarkably quickly as input size grows.
- Statistics and data science — factorial calculations underlie key statistical distributions and formulas used in hypothesis testing and probability modeling.
- Puzzle-solving and recreational mathematics — many classic counting puzzles (how many ways can a deck of cards be shuffled, or a group of people be seated) reduce directly to factorial calculations.
Frequently asked questions
What is 0! (zero factorial), and why does it equal 1? By mathematical convention, 0! is defined as exactly 1, not 0 — this might seem counterintuitive, but it's essential for combinatorics formulas (like permutations and combinations) to work correctly at their boundary cases, and it's consistent with the broader mathematical pattern where an "empty product" (multiplying together nothing at all) is conventionally defined as 1.
How large does a factorial get, really? Extremely large, extremely fast — 10! is 3,628,800, but 20! already exceeds 2.4 quintillion, and 100! is a number with 158 digits, far larger than the estimated number of atoms in the observable universe, illustrating just how explosively factorial growth accelerates compared to more familiar exponential growth.
Can you calculate the factorial of a negative number or a fraction? Not with the standard factorial definition, which only applies to non-negative whole numbers — however, a related, more advanced function called the Gamma function extends the concept of "factorial" to work with fractions and even most negative and complex numbers, a genuinely elegant generalization studied in more advanced mathematics.
Further reading
Wikipedia — Factorial — The mathematical definition, notation history and applications of factorials.