Combinatorial proofs of the identity $(a+b)^2 = a^2 +b^2 +2ab$

Question

The question I have is to give a combinatorial proof of the identity $(a+b)^2 = a^2 +b^2 +2ab$.

I understand the concept of combinatorial proofs but am having some trouble getting started with this problem, any help would be appreciated.

Answer 1

Draw a square of side $a+b$ and a line parallel to each pair of sides. Where should you place the line?

Imagine this broken into a checkerboard:

Answer 2

Hint. You have $a$ different blue shirts and $b$ different pink shirts. In how many ways can you choose one shirt to wear today and one to wear tomorrow?

Answer 3

Combinatorially argue that ${a+b \choose 2} = {a \choose 2} + {b \choose 2} + ab$