Basic Examples

No elements are in their original place:
[◼]
DerangementQ
@{3,1,4,2}
In[]:=
True
Out[]=
For even n, the list
{n,…,3,2,1}
is a derangement:
[◼]
DerangementQ
@Reverse@Range@12
In[]:=
True
Out[]=

Scope

The chances are about 1 in 3 that a permutation is a derangement:
Table
[◼]
DerangementQ
@PermutationList@RandomPermutation@12,{9}
In[]:=
{True,False,False,False,False,False,False,False,True}
Out[]=