Today I gave a combinatoric argument for the binomial theorem. For me this is the most elegant and clear way to understand it, but there are other proofs which rely on very different techniques.

There is a nice article here, discussing the contrast between three different proofs of this same theorem (using combinatorics, induction and calculus) and arguing why different measures of a beautiful proof would say that each of the three proofs has benefits over the others.

Which proof have you so far seen which you have found most appealing? Can you find a proof that we have already done and write it in a more aesthetically satisfying way?