Pascal's triangle and combinatorial proofs This recent question got me thinking, if a textbook (or an exam) tells a student to give a combinatorial proof of something involving (sums of) binomial coefficients, would it be enough to show that by Pascal's triangle these things do add up, or would you fail an answer like that? What if we didn't call it Pascal's triangle but "the number of paths that stop at some point at step $i$ during a one-dimensional random walk"?

I would argue that a combinatorial proof is something more substantial than pointing out a pattern in a picture!  If we are at the level of "combinatorics" then we are also at the level of proofs and as such, the phrase "combinatorial proof" asks for a proof but in the combinatorial (or counting) sense.
A proof by example, i.e. "this pattern holds in the small portion of Pascal's Triangle that I have drawn", is not a proof period, combinatorially or otherwise.  The general case of such a property could be verified combinatorially, but simply observing it would not constitute a combinatorial proof in itself.  At least that's the way I see it.
