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```        Hello frnds..........i m yogita....preparing for iit jee-2012

Can anybody give me the proof for dearrangement series in permutations..........??????```
7 years ago

SAGAR SINGH - IIT DELHI
879 Points
```										Dear yogita,
Any change in the existing order of things is called a derangement.
If 'n' things are arranged in a row,  the number of ways in which they can, be deranged so that none of them  occupies its original place is

and it is denoted by D(n).
A question on derangement can be of the following kind:
Illustration:
Supposing 4 letters are placed in 4  different envelopes. In how many ways can be they be taken out from  their original envelopes and distributed among the 4 different envelopes  so that no letter remains in its original envelope?
Solution:
Using the formula for the number of  derangements that are possible out of 4 letters in 4 envelopes, we get  the number of ways as :
4!(1 - 1 + 1/2! - 1/3! + 1/4!) = 24(1 - 1 + 1/2 - 1/6 + 1/24) = 9.

We are all  IITians and here to help you in your IIT  JEE preparation.  All the best.

Sagar Singh
B.Tech IIT Delhi

```
7 years ago
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