The problem of rearranging objects so that each object is assigned to a place different from its original place is referred to as the problem of dearrangments. Here, we need to find out the number of dearrangements possible with  letters and  envelopes.
Let us denote by  the number of dearrangements possible with  things. We will attempts a general solution, that is for an arbitrary  , and then substitute 
Denote by  the  letter and by  , the original envelope of the  letter.
Consider the envelope  . It can be assigned a wrong letter in  ways. Suppose that we assign the letter  to  .
The dearrangements that can now arise can be divided into two mutually exclusive classes:
| (i) Those in which is assigned to  |
| (ii) Those in which is not assigned to  |
If  is assigned to  , we have the following configuration:
In this case, we have a remaining of  letters which can be dearranged in  ways.
Suppose the other case now, where we do not assign  to  . In this case, we have  letters to dearrange (  will also be counted as a letter to be dearranged since it is being assigned to an envelope other than  ). This can be done in  ways.
Thus, if we give  to  , the total dearrangements possible are 
Since  can assigned a wrong letter in  ways, the overall total number of dearrangements  is
We have thus related the  order ‘dearrangements–coefficient’  to lower order coefficients.
We now have to somehow use this relation to obtain  only in terms of  . This is how we do it:
But if we substitute  we get
We substitute  back into  to get
It is obvious that  since one letter cannot be dearranged while  because two letters can be dearranged in only one way : by exchanging them.
Thus,
We still have not obtained a relation involving only  . We do it using  repeatedly
Now we substitute  successively and add the corresponding sides of all the relations so obtained to get
Since  , we finally get
This is the number of de arrangements possible with  things For  , we have
Thus, there are  ways to rearrange the letters back into their envelopes so that each letter goes to a wrong envelope.
Since  is a small number, we could have worked out an alternative solution as follows:
You are urged to work out the solution by this way yourself.
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