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.
2, 3, 7, 9, 10
1, 2, 3, 9
2, 4, 9, 10
2, 3, 9
2, 9, 10
2, 9
2,
9
is therefore equal to 
, corresponding to the seven elements 
.
