Josh, if you think about a picture like the one Eliezer drew (but in however many dimensions you like) it's kinda obvious that the leading term in the difference between two n-cubes consists of n (n-1)-cubes, one per dimension. So the leading term in the next difference is n(n-1) (n-2)-cubes, and so on. But that doesn't really give the n! thing *at a glance*. I'm not convinced that anything to do with nth differences can really be seen *at a glance* without some more symbolic reasoning intervening.

James Bach, I suspect that the *really* good mathematicians can't be had for cheap because doing mathematics is so important to them, and the *quite* good mathematicians can't be had for cheap because they've taken high-paying jobs in finance or software or other domains where a mathematical mind is useful. But maybe it depends on what you count as "cheap" and what fraction of the mathematician's time you want to take up with tutoring...

Isabel, I think perhaps differentiation really *is* easier in some sense than differencing, not least because the formulae are simpler. Maybe that stops being true if you take as your basic objects not n^k but n(n-1)...(n-k+1) or something, but it's hard to see the feeling that n^k is simpler than that as mere historical accident.