Why is it important to study combinatorics? I was having a discussion with my friend Sayan Mukherjee about why we need to study combinatorics which admittedly, is not our favorite subject because we see very less motivation for it (I am not saying that there does not exist motivation for studying it, it's just that I have not found it).
Here are some of the "uses" of combinatorics that we could come up with:


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*Counting - the number of ways in which we can perform a finite sequence of operations and how objects can be arranged or selected. For example,the number of ways in which we can select $k$ odd and even elements from the set  $S=\{1,2,\dots, 2n\}$ so that at most 3 odd elements consecutive elements could occur in the section.

*Drawing bijections- The classic Stars and bars problem provides us key ideas to count the number of integral solutions to equations of the form $x_1+x_2+\dots x_n=k$.

*The Seven Bridges of Königsberg which captivated me as a child.
I have refrained from mentioning recursions and generating functions as I see them more as tools.
But I am looking for more motivation; counting, as described in problems seems to be tip of the iceberg and I will appreciate more examples where combinatorics and graph theory can be powerful tools. Can we please have a list of uses of combinatorics? I am not looking for applications to industry, just pure math.
It is  not essential that the answers be pitched at high-school level; additional info will certainly be fun to revisit!

Combinatorics turns out to be pretty important for group theory and probability. It's also relevant for topology, analysis, etc.
