By the wording, if even 1 letter is in the correct envelope, it counts as a fail.

In all, there are 4!=(4*3*2*1) or 24 possible ways to put the four letters into 4 different envelopes. Now we need to subtract the failures.

Failure will occur if exactly 1 letter is in the correct envelope. Since there are 4 envelops, there are 4 different ways to accomplish this. So, the value is 4 chose 1. Likewise, there are 4 choose 2 ways to insert 2 letters into the correct envelopes, 4 choose 3 ways to insert 3 letters into the 4 envelopes, and 4 chose 4 ways to insert all 4 letters into the correct envelopes.

Let n=4 be the number of envelopes in which at least 1 of n letters are to be incorrectly placed and let m be the total number of ways of doing so. Then, m can be expressed as below.

m=24−[(4C1)+(4C2)+(4C3)+(4C4)]

m=24−[4+6+4+1]=24−15=9

So, there are 9 ways in which the 4 letters can be inserted into the 4 different envelops in which at least 1 is incorrect.