Algebraic Proof of Combinatorial Identity

[Shoney45]

Homework Statement

Verify C(n,k) = C(n-1,k) + C(n-1,k-1) algebraically.

Homework Equations

N/A

The Attempt at a Solution

I've set the identity up factorially like so: $$(n-1)!/k!(n-1-k)! + (n-1)!/[(k-1)!(n-2-k)!$$

I'm having a really hard time getting started here. That is the story in a nutshell. I've been staring at this for a while, and just can't get started. I need a nudge getting the ball rolling a little bit.

 

[Hurkyl]

There are a lot of factors in common to both terms...