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

Can anybody give me the proof for dearrangement series in permutations..........??????

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.

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All the best.

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