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In the following two lists, no element appears in its original place:

[◼]	Derangements

@3

In[]:=

{{2,3,1},{3,1,2}}

Out[]=

Here is the next case, condensed:

Row/@

[◼]	Derangements

@4

In[]:=

{2143,2341,2413,3142,3412,3421,4123,4312,4321}

Out[]=

The number of lists grows with

n!/e

:

7!/E-Length@

[◼]	Derangements

@Range@7//N

In[]:=

0.112384

Out[]=

In fact the number of derangements is given by the built-in subfactorial function:

Subfactorial@7Length@

[◼]	Derangements

@Range@7

In[]:=

True

Out[]=