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

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.

• Are you supposing $a, b$ are positive integers? Feb 20, 2014 at 0:18
• no it just says, assume they are integers, not positive ones though Feb 20, 2014 at 0:30

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?

• so would you have a*b possibilities of what shirt to wear today? Feb 20, 2014 at 0:29
• Only if you are wearing two shirts at the same time ;-) Can you explain what multiplication means in combinatorics problems? And what addition means? Feb 20, 2014 at 0:33
• ok the comments are deleted, so would it be a+b ways that you could wear a shirt on a given day? Feb 20, 2014 at 0:41
• That's right, so now if you have to also make a choice for the second day...? Feb 20, 2014 at 0:43
• so would that imply that we can't use the shirt we used today? so would it be a+b-1 Feb 20, 2014 at 0:45

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:

• I would call this a geometric proof. Feb 20, 2014 at 0:19
• would you draw two lines through the middle of the square so that they were parallel to each of the sides? Feb 20, 2014 at 0:23
• @MaMing: yes, but if you do it on a checkerboard it becomes combinatorial. Feb 20, 2014 at 0:26
• @user130096: maybe they don't go through the middle. If they were $a$ from one side, for example. Feb 20, 2014 at 0:27
• A combinatorial proof is usually understood as one that establishes equality by counting the same group of things in two different ways. Here the whole square is $(a+b)^2$ and the four regions are $a^2,b^2,ab,ab$ Feb 20, 2014 at 0:44

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