2
$\begingroup$

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.

$\endgroup$
2
  • $\begingroup$ Are you supposing $a, b$ are positive integers? $\endgroup$
    – Ma Ming
    Feb 20, 2014 at 0:18
  • $\begingroup$ no it just says, assume they are integers, not positive ones though $\endgroup$
    – user130096
    Feb 20, 2014 at 0:30

3 Answers 3

7
$\begingroup$

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?

$\endgroup$
7
  • $\begingroup$ so would you have a*b possibilities of what shirt to wear today? $\endgroup$
    – user130096
    Feb 20, 2014 at 0:29
  • $\begingroup$ Only if you are wearing two shirts at the same time ;-) Can you explain what multiplication means in combinatorics problems? And what addition means? $\endgroup$
    – David
    Feb 20, 2014 at 0:33
  • $\begingroup$ ok the comments are deleted, so would it be a+b ways that you could wear a shirt on a given day? $\endgroup$
    – user130096
    Feb 20, 2014 at 0:41
  • $\begingroup$ That's right, so now if you have to also make a choice for the second day...? $\endgroup$
    – David
    Feb 20, 2014 at 0:43
  • $\begingroup$ so would that imply that we can't use the shirt we used today? so would it be a+b-1 $\endgroup$
    – user130096
    Feb 20, 2014 at 0:45
2
$\begingroup$

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:

enter image description here

$\endgroup$
10
  • 8
    $\begingroup$ I would call this a geometric proof. $\endgroup$
    – Ma Ming
    Feb 20, 2014 at 0:19
  • $\begingroup$ would you draw two lines through the middle of the square so that they were parallel to each of the sides? $\endgroup$
    – user130096
    Feb 20, 2014 at 0:23
  • 1
    $\begingroup$ @MaMing: yes, but if you do it on a checkerboard it becomes combinatorial. $\endgroup$ Feb 20, 2014 at 0:26
  • $\begingroup$ @user130096: maybe they don't go through the middle. If they were $a$ from one side, for example. $\endgroup$ Feb 20, 2014 at 0:27
  • 1
    $\begingroup$ 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 $a^2,b^2,ab,ab$ $\endgroup$ Feb 20, 2014 at 0:44
1
$\begingroup$

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

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .