Pascal's Rule to Choose Function

[manymanyrules]

Could someone please explain, or at least point me in the right direction, of how you start with Pascal's rule and end up with the choose function? So far, everwhere I have looked, there is a kind of hand waiving over it and then a proof that starts with the choose function shows how it acts to Pascal's rule.

To clarify things, by Pascal's rule I mean for a function f(n,p):
f(n,p) = f(n-1, p) + f(n-1, p-1)

and by choose function I mean:
C(n,p) = n! / ( p!*(n-p)! )

thanks