Question

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
A. `D_(n)-nD_(n-1)=(-1)^(n)`
B. `D_(n)-(n-1)D_(n-1)=(-1)^(n-1)`
C. `D_(n)-nD_(n-1)=(-1)^(n-1)`
D. `D_(n)-D_(n-1)=(-1)^(n-1)`

Answer 1

Correct Answer - A
`(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)`