Exact Derangement Number Calculator

Use !0 = 1, !1 = 0, and multiply the sum of the two prior derangement counts by n minus one. The empty arrangement contributes one base case, while one object cannot be deranged. Inputs remain whole numbers inside the displayed domain, and the page never rounds a decimal into an accepted integer. This keeps the reported result tied to one explicit mathematical definition rather than an unstated convention.

Objects and positions are distinct, and every object must move; this is stricter than merely changing the overall order. Enumerate a very small n or substitute the two preceding recurrence values. Reuse the displayed recurrence, congruence, or counting identity with the same inputs. The verification evidence checks this calculator's result, but it does not establish assumptions or conclusions outside the stated mathematical domain.

The count is capped at n = 18 so both factorial and derangement results remain exact safe integers. Larger factorial-scale counts require arbitrary-precision integer output. A finite limit prevents browser stalls and avoids presenting an unsafe floating-point integer as exact. Work beyond that limit belongs in arbitrary-precision software with explicit resource controls and independent validation.