What a factorial is and where it shows up
A plain guide to n!, why it grows so fast, how it powers permutations and combinations, and why big-number precision matters.
The definition, one multiplication at a time
A factorial is a running product. To find n! you start at 1 and multiply in every whole number up to and including n, so 4! is 1 times 2 times 3 times 4, which is 24. Each factorial contains the one before it, meaning n! equals n times (n minus 1)!, which is why 5! is simply 5 times 24, or 120. That recursive shape is the reason a small increase in n causes a large jump in the answer.
Why the numbers explode
Because every step multiplies rather than adds, factorials grow faster than exponential functions. 10! is already 3,628,800, and 20! is 2,432,902,008,176,640,000, a 19 digit number. By 100! the result has 158 digits, well beyond what a standard 64 bit number can hold without losing accuracy. This calculator sidesteps that limit by computing with big integers, so the last digit is as trustworthy as the first.
Counting arrangements and choices
The most direct use of a factorial is counting orderings: there are n! ways to arrange n distinct objects in a line, so five books can sit on a shelf in 120 different orders. Factorials also build the formulas for permutations and combinations, where choosing and ordering items is expressed as ratios of factorials. That is why a deck of 52 cards has 52! possible shuffles, a number so vast it dwarfs the count of atoms on Earth.
Factorials beyond counting
The idea reaches past simple counting into calculus and analysis. Factorials sit in the denominators of the Taylor series that define functions like the exponential, sine and cosine, which is how those series converge so quickly. For inputs that are not whole numbers, mathematicians use the gamma function, a smooth extension of the factorial that this tool does not calculate. Within the whole-number range it is built for, though, the exact result here matches the strict definition every time.