Factorial

n! counts the orderings of n distinct things: the first slot has n candidates, the next n − 1, and so on. Five books on a shelf can be arranged 5! = 5 × 4 × 3 × 2 × 1 = 120 ways; ten books already reach 10! = 3,628,800. The factorial's explosive growth is why brute-force scheduling and travelling-salesman searches collapse so quickly: 52!, the number of orders a shuffled deck can take, is about 8 × 10⁶⁷, so essentially every properly shuffled deck in history has been a first.

The "!" notation is Christian Kramp's, introduced in 1808 mainly because printers hated the older bracket symbols. Two conventions catch people out: 0! = 1, not 0 (there is exactly one way to arrange nothing, and the whole edifice of combinations depends on it) and n! is only defined for whole numbers in counting problems. This calculator quietly extends to fractional inputs through the gamma function (Euler's continuous version, with n! = Γ(n + 1)), so 0.5! returns $\sqrt{\pi}/2 \approx 0.886$, but for counting keep n a whole number. Above 170 the answer exceeds what a double can hold.