Factorial

The factorial of a non-negative integer n, written n!, is the product of every positive integer from 1 up to n. By convention, 0! is defined to equal 1, which keeps many formulas consistent and avoids awkward special cases. Factorials grow astonishingly fast: even a modest input like 20 already yields a number with nineteen digits, far outpacing exponential growth. They are the natural way to count arrangements, since n! gives the number of distinct orders in which n objects can be lined up. This makes the factorial a cornerstone of combinatorics, where it underlies the formulas for permutations and combinations. It also appears throughout probability, algebra, and calculus, from the binomial theorem to the power-series expansions of functions like the exponential, sine, and cosine. Whenever you need to know "how many ways" something can happen, the factorial is usually close at hand.