Home
Class 12
MATHS
Let theta=(a(1),a(2),a(3),...,a(n)) be a...

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)`

Text Solution

Verified by Experts

The correct Answer is:
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)`
Promotional Banner

Topper's Solved these Questions

  • PERMUTATION AND COMBINATION

    CENGAGE|Exercise Question Bank|4 Videos

  • PERMUTATION AND COMBINATION

    CENGAGE|Exercise Multiple Correct Answer|2 Videos

  • PARABOLA

    CENGAGE|Exercise Question Bank|9 Videos

  • PRINCIPLE OF MATHEMATICAL INDUCTION

    CENGAGE|Exercise Exercise|9 Videos

Similar Questions

Explore conceptually related problems

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 . D_(n) is equal to

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 . There are 5 different colour balls and 5 boxes of colours same as those of the balls. The number of ways in which one can place the balls into the boxes, one each in a box, so that no ball goes to a box of its own colour is

If A_(1), A_(2),..,A_(n) are any n events, then

Find the indicated terms of the sequences whose nth terms are given by a_(n)=-(n^(2)-4), a_(4) and a_(11)

If a_(1),a_(2)a_(3),….,a_(15) are in A.P and a_(1)+a_(8)+a_(15)=15 , then a_(2)+a_(3)+a_(8)+a_(13)+a_(14) is equal to

If the sequence a_(1),a_(2),a_(3) ,…….. Is an A.P., then the sequence a_(5), a_(10), a_(15) ,…….. Is ………..

Let a_(1),a_(2),a_(3), . . . be a harmonic progression with a_(1)=5anda_(20)=25 . The least positive integer n for which a_(n)lt0 , is

Find the indicated terms of the sequences whose nth terms are given by a_(n)=(5n)/(n+2), a_(6) and a_(13)

Let A_(0) be the area enclosed by the orbit in a hydrogen atom .The graph of ln (A_(0) //A_(1)) against ln(n)