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.