Question
A derangement of n distinct items is a permutation in which no item appears in its original position. Given n (0 <= n <= 10^6) and a modulus mod (>= 2), return the number of derangements of n items, modulo mod. By convention there is exactly one derangement of 0 items (the empty arrangement) and zero derangements of 1 item. For example, 3 items have 2 derangements.
count_derangements(n: int, mod: int) → int[3,1000000007]out2State your approach and its time/space complexity out loud before you optimize. Handle the edge cases (empty input, duplicates, overflow), and say why you chose this over the brute force. Green tests are the floor, not the grade.
Vibe coding: describe the solution in plain language (or narrate it) and the coach grades your approach. Generating runnable code from your description is coming next.