Editing Inclusion–exclusion principle (section)

==A special case==

The situation that appears in the derangement example above occurs often enough to merit special attention.<ref>{{harvnb|Brualdi|2010|loc=pp. 167–8}}</ref> Namely, when the size of the intersection sets appearing in the formulas for the principle of inclusion–exclusion depend only on the number of sets in the intersections and not on which sets appear. More formally, if the intersection

:<math>A_J:=\bigcap_{j\in J} A_j</math>

has the same cardinality, say ''α<sub>k</sub>'' = |''A<sub>J</sub>''|, for every ''k''-element subset ''J'' of {1,&nbsp;...,&nbsp;''n''}, then

:<math>\left |\bigcup_{i=1}^n A_i\right|  =\sum_{k=1}^n (-1)^{k-1}\binom nk \alpha_k.</math>

Or, in the complementary form, where the universal set ''S'' has cardinality ''α''<sub>0</sub>,

:<math>\left |S \smallsetminus \bigcup_{i=1}^n A_i\right|  =\alpha_0 - \sum_{k=0}^n (-1)^{k-1}\binom nk \alpha_k.</math>