Derangement, example, paradox? How can we explain that $!0 =1 $, but $!1=0$?
I understand the case of permutations. I get why $0! =1$, and that $1!$ is also $1$.
This result doesn't argue with my intuition.
But, when it comes to derangements, it's hard to understand for me. Please help me.

$0!$ is one, since the number of bijections between $\emptyset$ and $\emptyset$ is $1$. moreover the one bijection that exists does not send any element of $\emptyset$ to itself, so it is also a derrangement and we conclude $!0=1$
The number of bijections from $\{1\}$ to $\{1\}$ is also $1$, so $1!=1$. However this bijection sends $1$ to itself, and therefore is not a derrangement. The number of derrangements on the set $\{1\}$ is therefore $0$, so $!1=0$.
