This formula can be generalized in the following beautiful manner. Let be a p-system
of
consisting of sets ,
..., , then
(3)
where the sums are taken over k-subsets of . This formula holds for infinite sets as well as finite sets (Comtet 1974, p. 177).
The principle of inclusion-exclusion was used by Nicholas Bernoulli to solve the recontres problem of finding the number of derangements
(Bhatnagar 1995, p. 8).
For example, for the three subsets ,, and of ,
the following table summarizes the terms appearing the sum.
#
term
set
length
1
{2, 3, 7, 9, 10}
5
{1, 2, 3, 9}
4
{2, 4, 9, 10}
4
2
{2, 3, 9}
3
{2, 9, 10}
3
{2, 9}
2
3
{2,
9}
2
is therefore equal to ,
corresponding to the seven elements .
denote the cardinal number of set 
, then it follows immediately that
denotes union, and 
denotes intersection.
The more general statement
be a p-system
of 
consisting of sets 
,
..., 
, then
. This formula holds for infinite sets 
as well as finite sets (Comtet 1974, p. 177).
, 
, and 
of 
,
the following table summarizes the terms appearing the sum.
is therefore equal to 
,
corresponding to the seven elements 
.
Extensions." PhD thesis. Columbus, OH: Ohio State
University, 1995.Comtet, L.