Discrete Math?

Question

How many ordered quadruples (w,x,y,z) of positive integers are there with w+x+y+z=40

Answer 1

Think of them as the lengths of four different pieces of a line segment that's 40 units long. Now instead of looking for the the lengths, let's (equivalently) look for the interior points.

Then you're looking for three integers a < b < c, all in [1,39]. So take 39-choose-3 and you're done!

Really, the only tricky part is convincing yourself that you don't have to divide or multiply by an additional factor of 3!