Which of the following is not an application of inclusion-exclusion principle? ans. Graph coloring Matching of bipartite graphs Maximum flow problem Counting intersections

The Inclusion-Exclusion Principle is a fundamental concept in combinatorics - the study of counting and arranging objects. It is used to calculate the number of elements in the union of multiple sets, where some elements may be counted more than once.

1. Graph coloring: This is not an application of the Inclusion-Exclusion Principle. Graph coloring is a way of assigning colors to the vertices of a graph in such a way that no two adjacent vertices share the same color. This is a problem in graph theory, not combinatorics.

2. Matching of bipartite graphs: This is also not an application of the Inclusion-Exclusion Principle. This is a problem in graph theory where we try to find a matching in a bipartite graph that covers all vertices. The Inclusion-Exclusion Principle does not apply here.

3. Maximum flow problem: This is not an application of the Inclusion-Exclusion Principle. The maximum flow problem is a network flow problem where we try to maximize the amount of flow from a source to a sink in a flow network. This is a problem in optimization, not combinatorics.

4. Counting intersections: This is an application of the Inclusion-Exclusion Principle. When we have multiple sets and we want to count the number of elements in their union, we can use the Inclusion-Exclusion Principle to avoid double-counting elements that appear in more than one set.

So, the answer is that the Inclusion-Exclusion Principle is not applied in Graph coloring, Matching of bipartite graphs, and Maximum flow problem.