TITLE: Why is it important to study combinatorics?
QUESTION [36 upvotes]: 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:

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!

REPLY [7 votes]: Another important application of combinatorics is in representation theory, symmetric functions, and the study of varieties with lots of symmetries (Grassmannians, flag varieties, toric varieties, symmetric varieties, spherical varieties).
In general, objects with enough symmetries can often be described by discrete (often finite) data.  Combinatorics comes into play in order to parameterize the data and, more generally, because relationships between objects are often described in terms of combinatorics of the data.
As a simple example, suppose you want to study $k$-dimensional subspaces of an $n$-dimensional vector space $V$.  One way to understand such a subspace $W$ is to consider a "flag" of $V$, which is a sequence of subspaces $$0 = V_0 \subset V_1 \subset \dots \subset V_i  \subset \dots \subset V_n = V,$$ with $\dim(V_i) = i$.  You can divide the different subspaces $W$ based on the weakly increasing sequence of numbers $$0 = \dim(W \cap V_0) \leq \dim(W \cap V_1) \leq \dots \leq \dim(W \cap V_i) \leq \dots \leq \dim(W \cap V_n) = k,$$
with the restriction that at each stage the dimension can go only up 1 or stay the same.  An equivalent way to describe the position of $W$ relative to the flag is to give a sequence of length $n$ consisting of $k$ 1's and $n-k$ 0's, with a $1$ occurring in the $i$-th position exactly when $\dim(W \cap V_i) > \dim(W \cap V_{i-1})$.
A simple observation is that the number of different "cells" that you have divided the subspaces into is equal to $n \choose k$, since this counts the number of such strings of 0's and 1's.  This is only the beginning of a rich interplay between combinatorics and geometry.  The moral is that if you want to study an object with symmetries, you are very likely to encounter some interesting combinatorics along the way.