Why combinatorics is neat

[gusl]

Because it allows you to reason outside of regular mathematical domain.
By extracting the "meaning" of different concepts and combining them, you can derive theorems of arithmetic. I suppose this is not too different from the concepts where all mathematical reasoning came, but here we have a parallel system of reasoning consistent with the existing system.

I suppose it's analogous to reasoning with graphs, diagrams or other visual devices.


CONCEPT : ARITHMETIC EXPRESSION

# of binary strings of length n with 0 "1"s: C(0,n)
+
# of binary strings of length n with 1 "1"s: C(1,n)
+
# of binary strings of length n with 2 "1"s: C(2,n)
+
# of binary strings of length n with 3 "1"s: C(3,n)
+
.
.
.
+
# of binary strings of length n with n-1 "1"s: C(n-1,n)
+
# of binary strings of length n with n "1"s: C(n,n)

=

# of binary strings of length n: 2^n


So we have SUM_i(C(i,n)) = 2^n
because we have the concept that the sets with constant numbers of "1"s are a partition of the large set (of size 2^n)

The equation of CONCEPTS entails an equation of ARITHMETIC EXPRESSIONS.

This theorem can be proven using the axioms of arithmetic, or it can be proven using the reasoning I just described.
The point is that there exist "common sense" tools for mathematical reasoning which are not formally axiomatized. Is this related to diagrammatic reasoning?

There are a bunch of examples like this in combinatorics.

For similar examples of combinatorial identities, look at the many properties of Pascal's triangle.