Number of Derangement Define i=k as Property Ak,and Ak is used for the subset of all permutations satisfying property Ak. The number of derangemen t is: N(AA2A.An)=N-S+S2-S3+.+(-1)S.++(-1)”S where N=n! where S is 144.4 1≤h≤i2≤ik≤n Note:S is the number of permutatio ns keeping exactly k elements in their original positions,and the other n-kelements as any possible permutatio n.So: s0s8-y--是Number of Derangement ◼ Define ik=k as Property Ak , and Ak is used for the subset of all permutations satisfying property Ak . ! ! ( 2)!;..., ( )! 2 ( 1)!; 1 possible permutatio n. So : in their original positions, and the other elements as any Note : is the number of permutatio ns keeping exactly elements where is | ... | where ! ( ... ) ... ( 1) ... ( 1) The number of derangemen t is 1 2 1 ... 1 2 3 1 2 3 1 2 1 2 k n n k k n n S n n S n S n k S k S A A A N n N A A A A N S S S S S k k i i i n k i i i n n k k n k k − =         − =         − =         = −    = = − + − + + − + + −     ：