Editing Derangement (section)

== Example ==
[[File:Derangement4.png|thumb|right|The 9 derangements (from 24 permutations) are highlighted.]]
Suppose that a professor gave a test to 4 students – A, B, C, and D – and wants to let them grade each other's tests. Of course, no student should grade their own test. How many ways could the professor hand the tests back to the students for grading, such that no student receives their own test back? Out of [[v:Symmetric group S4#tables|24 possible permutations]] (4!) for handing back the tests,
:{| style="font:125% monospace;line-height:1;border-collapse:collapse;"
|<span style="color:red;font-weight:bold;">ABCD</span>,
|<span style="color:red;font-weight:bold;">AB</span>DC,
|<span style="color:red;font-weight:bold;">A</span>CB<span style="color:red;font-weight:bold;">D</span>,
|<span style="color:red;font-weight:bold;">A</span>CDB,
|<span style="color:red;font-weight:bold;">A</span>DBC,
|<span style="color:red;font-weight:bold;">A</span>D<span style="color:red;font-weight:bold;">C</span>B,
|-
|BA<span style="color:red;font-weight:bold;">CD</span>,
|<span style="color:deepskyblue;font-style:italic;">BADC</span>,
|BCA<span style="color:red;font-weight:bold;">D</span>,
|<span style="color:deepskyblue;font-style:italic;">BCDA</span>,
|<span style="color:deepskyblue;font-style:italic;">BDAC</span>,
|BD<span style="color:red;font-weight:bold;">C</span>A,
|-
|CAB<span style="color:red;font-weight:bold;">D</span>,
|<span style="color:deepskyblue;font-style:italic;">CADB</span>,
|C<span style="color:red;font-weight:bold;">B</span>A<span style="color:red;font-weight:bold;">D</span>,
|C<span style="color:red;font-weight:bold;">B</span>DA,
|<span style="color:deepskyblue;font-style:italic;">CDAB</span>,
|<span style="color:deepskyblue;font-style:italic;">CDBA</span>,
|-
|<span style="color:deepskyblue;font-style:italic;">DABC</span>,
|DA<span style="color:red;font-weight:bold;">C</span>B,
|D<span style="color:red;font-weight:bold;">B</span>AC,
|D<span style="color:red;font-weight:bold;">BC</span>A,
|<span style="color:deepskyblue;font-style:italic;">DCAB</span>,
|<span style="color:deepskyblue;font-style:italic;">DCBA</span>.
|}
there are only 9 derangements (shown in blue italics above). In every other permutation of this 4-member set, at least one student gets their own test back (shown in bold red).

Another version of the problem arises when we ask for the number of ways ''n'' letters, each addressed to a different person, can be placed in ''n'' pre-addressed envelopes so that no letter appears in the correctly addressed envelope.