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.
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Martin Sleziak
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user130096
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Are you supposing $a, b$ are positive integers? – Ma Ming Feb 20 '14 at 00:18
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no it just says, assume they are integers, not positive ones though – user130096 Feb 20 '14 at 00:30
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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?
David
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Only if you are wearing two shirts at the same time ;-) Can you explain what multiplication means in combinatorics problems? And what addition means? – David Feb 20 '14 at 00:33
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ok the comments are deleted, so would it be a+b ways that you could wear a shirt on a given day? – user130096 Feb 20 '14 at 00:41
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That's right, so now if you have to also make a choice for the second day...? – David Feb 20 '14 at 00:43
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so would that imply that we can't use the shirt we used today? so would it be a+b-1 – user130096 Feb 20 '14 at 00:45
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Maybe you can wear the same shirt again, maybe not. I think I should leave you to figure out which assumption is going to solve your problem. – David Feb 20 '14 at 00:50
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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:
Ross Millikan
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would you draw two lines through the middle of the square so that they were parallel to each of the sides? – user130096 Feb 20 '14 at 00:23
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1@MaMing: yes, but if you do it on a checkerboard it becomes combinatorial. – Ross Millikan Feb 20 '14 at 00:26
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@user130096: maybe they don't go through the middle. If they were $a$ from one side, for example. – Ross Millikan Feb 20 '14 at 00:27
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well it is a combinatorial proof so wouldn't that be a good thing if it became a combinatorial? – user130096 Feb 20 '14 at 00:28
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@RossMillikan where would the lines go then? it makes sense for each line to go through the middle doesn't it? – user130096 Feb 20 '14 at 00:33
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For example, let $a=5,b=3$ and think of a checkerboard ($8 \times 8)$ If you draw a line with five columns to the left of it and another with five rows in front of it, you divide the checkerboard into four regions. What are the areas of each region? – Ross Millikan Feb 20 '14 at 00:35
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@RossMillikan: I'm just being curious. What is the essential difference between a proof that involves counting and a combinatorial proof in general ? – r9m Feb 20 '14 at 00:39
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1A 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 $a^2,b^2,ab,ab$ – Ross Millikan Feb 20 '14 at 00:44
