factorial
The factorial of a positive whole number n, written n! and read “n factorial,” is the product of every positive integer from 1 up to n. So 5! = 5·4·3·2·1 = 120. It is the answer to “in how many different orders can I line up n distinct objects?”
Factorials grow astonishingly fast — far quicker than any power — which is why even 13! already exceeds a billion. They satisfy the simple recursion n! = n·(n − 1)!, so each factorial is just the next integer times the one before it.
By a useful convention, 0! = 1. That is not an oversight: there is exactly one way to arrange nothing, and the value 1 keeps formulas like the binomial coefficient C(n, k) = n!/(k!(n−k)!) working at the edges. Factorials are defined only for non-negative integers; there is no “2.5!” in elementary algebra.
4! = 4·3·2·1 = 24, the number of ways to arrange 4 books in a row.
n! counts the orderings of n distinct objects.
0! = 1 by definition — a fact people often misremember as 0. It makes the empty-arrangement count and many formulas come out right.