InclusionExclusion and its various Applications
In the field of Combinatorics, it is a counting method used to compute the cardinality of the union set. According to basic InclusionExclusion principle:
 For 2 finite sets and , which are subsets of Universal set, then and are disjoint sets.

Hence it can be said that,
.
 Similarly for 3 finite sets , and ,
Principle :
InclusionExclusion principle says that for any number of finite sets , Union of the sets is given by = Sum of sizes of all single sets – Sum of all 2set intersections + Sum of all the 3set intersections – Sum of all 4set intersections .. + Sum of all the iset intersections.
In general it can be said that,
Properties :
 Computes the total number of elements that satisfy at least one of several properties.
 It prevents the problem of double counting.
Example 1:
As shown in the diagram, 3 finite sets A, B and C with their corresponding values are given. Compute .
Solution :
The values of the corresponding regions, as can be noted from the diagram are –
By applying InclusionExclusion principle,
Applications :
 Derangements
To determine the number of derangements( or permutations) of n objects such that no object is in its original position (like Hatcheck problem).
As an example we can consider the derangements of the number in the following cases:
For i = 1, the total number of derangements is 0.
For i = 2, the total number of derangements is 1. This is .
For i = 3, the total number of derangements is 2. These are and 3 1 2.