Editing Inclusion–exclusion principle (section)

===Permutations with forbidden positions===
A [[permutation]] of the set ''S'' = {1, ..., ''n''} where each element of ''S'' is restricted to not being in certain positions (here the permutation is considered as an ordering of the elements of ''S'') is called a ''permutation with forbidden positions''. For example, with ''S'' = {1,2,3,4}, the permutations with the restriction that the element 1 can not be in positions 1 or 3, and the element 2 can not be in position 4 are: 2134, 2143, 3124, 4123, 2341, 2431, 3241, 3421, 4231 and 4321. By letting ''A<sub>i</sub>'' be the set of positions that the element ''i'' is not allowed to be in, and the property ''P''<sub>''i''</sub> to be the property that a permutation puts element ''i'' into a position in ''A<sub>i</sub>'', the principle of inclusion–exclusion can be used to count the number of permutations which satisfy all the restrictions.<ref>{{harvnb|Brualdi|2010|loc=pp. 177–81}}</ref>

In the given example, there are 12 = 2(3!) permutations with property ''P''<sub>1</sub>, 6 = 3! permutations with property ''P''<sub>2</sub> and no permutations have properties ''P''<sub>3</sub> or ''P''<sub>4</sub> as there are no restrictions for these two elements. The number of permutations satisfying the restrictions is thus:

:4! − (12 + 6 + 0 + 0) + (4) = 24 − 18 + 4 = 10.

The final 4 in this computation is the number of permutations having both properties ''P''<sub>1</sub> and ''P''<sub>2</sub>. There are no other non-zero contributions to the formula.