A few decades ago my algebra teacher showed how to construct the expression for binomial coefficients. If I start with Pascal's recursion, and propose C(n,k)=n!/k!(n-k)!, I can prove it to be so through induction. But that doesn't give me that happy feeling that comes with understanding. It can't be that hard; so I'm feeling really dumb at this point. How does one build up the relationship C(n,k)=n!/k!(n-k)! from scratch?