For positive integers n, r show that:

C(n+r+1,r) = C(n+r,r) + C(n+r-1,r-1) + ... + C(n+2,2) + C(n+1,1) + C(n,0)

= C(n+r,n) + C(n+r-1,n) + ... + C(n+2,n) + C(n+1,n) + C(n,n)

Now, I've tried expanding the combinations, and I was able to get (n!) in the denominator of every term, though I'm not sure if that is even relevant to the proof. Not far, I know, but essentially, I'm stuck there and I don't know where to go.

Thanks in advance for any and all help.