TITLE: What's Combinatorial Proof/Object/etc.?
QUESTION [6 upvotes]: In high school, when we talked about "combinatorics," we solely meant "mathematics of choice." For instance:

There are 10 people who want to sit around a table. In how many ways is this possible?
We have 50 balls and 20 boxes. In how many ways can we distribute balls into the boxes?
There are 10 apples and 5 oranges. In how many ways can we select 7 fruits?
...

I did have this certain wrong view of combinatorics unless I read about combinatorial objects (lists, sets, graphs, etc.) and combinatorial proofs. I tried reading several sources (Wikipedia, books, papers), but still don't have a clear understanding of when something is called combinatorial.

Could you please elaborate it?

REPLY [2 votes]: A combinatorial argument often consists in giving a bijection between two sets or at least has such an observation as its key step. More precisely, one often starts with one set $A$ whose elements one wishes to count. You then find a bijection from $A$ to some other set $B$ whose elements are easily counted, leading to some expression for the number of elements of $A$.
A very simple example would be the problem of counting the number of subsets of some finite set of size $n$; call this set of subsets $A$. We first notice there's a bijection from $A$ to the set $B$ of length $n$ binary strings. We can easily count the number of elements of $B$; it's just $2^n$. Hence $A$ also has $2^n$ elements.
More generally, a combinatorial argument proceeds by noticing that some statement to be proved is equivalent to the assertion that two sets are in bijection and then by constructing an explicit bijection.