Let theta=(a_(1),a_(2),a_(3),...,a_(n)) be a given arrangement of n distinct objects a_(1),a_(2),a_(3),…,a_(n). A derangement of theta is an arrangment of these n objects in which none of the objects occupies its original position. Let D_(n) be the number of derangements of the permutations theta. The relation between D_(n) and D_(n-1) is given by

Let theta=(a_(1),a_(2),a_(3),...,a_(n)) be a given arrangement of n di

1.	Let theta=(a_(1),a_(2),a_(3),...,a_(n)) be a given arrangement of n distinct objects a_(1),a_(2),a_(3),…,a_(n). A derangement of theta is an arrangment of these n objects in which none of the objects occupies its original position. Let D_(n) be the number of derangements of the permutations theta. The relation between D_(n) and D_(n-1) is given by
Answer» `D_(n)-nD_(n-1)=(-1)^(n)` `D_(n)-(n-1)D_(n-1)=(-1)^(n-1)` `D_(n)-nD_(n-1)=(-1)^(n-1)` `D_(n)-D_(n-1)=(-1)^(n-1)` Solution :`(a)` `D_(n)-nD_(n-1)=(-1)(D_(n-1)-(n-1)D_(n-2))` By implied induction on `n`, we OBTAIN `D_(n)-nD_(n-1)=(-1)^(n-2)(D_(2)-2D_(1))`, Where `D_(1)=0` and `D_(2)=1` `=(-1)^(n)`

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