There are $n$ letters and $n$ addressed envelopes. If the letters are placed in the envelopes at random, what is the probability that all the letters are not placed in the right envelope?
The number of ways in which no letter goes to the desired envelope is given by the $n$-th derangement number $D_n$. So the required probability is $\frac{D_n}{n!}$.
However, all the sources say that the answer is $1-\frac{1}{n!}$
I believe that the answer given by all these websites is incorrect since they assumed the complement to be the number of ways in which all letters go to the desired envelopes instead of the number of ways in which at least one letter goes to the correct envelope.
I would like someone to verify this since im outnumbered.