Learn about the fascinating concept of subfactorial (!n) in this 24-minute mathematics video that explores derangements and permutations without fixed points. Discover how this mathematical concept relates to real-world scenarios through an engaging example involving men, their hats, and divine intervention. Master the inclusion-exclusion principle to derive expressions for derangements and calculate the probability of random permutations being derangements. Progress through key topics including treacherous situations, notation fundamentals, argument construction, and probability calculations, culminating in an unexpected mathematical revelation.