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A binomial identity

mark

Show that

{{m+n}\choose{2}} - {m\choose2} - {n\choose2} = mn

using both combinatorial and algebraic techniques.

This problem is right out of the book, by the way.

asmith42

Here are the algebraic and combinatorial proofs. I’m not certain that my answer for the combinatorial proof is what you’re looking for, but I gave it my best shot.