How Do You Count Derangements?

Prerequisites: (This will be updated soon!)

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[hasunamarasekara]

Looks like there's a mistake at 1:49 since set {2, 1, 3} isn't a derangement because 3 is never moved from it's original position. The only derangements left not crossed out should instead be {2, 3, 1} and {3, 1, 2}.