Binomial coefficient proof - where have I gone wrong in my working?

Hi all,

I have been given the definition for the binomial coefficient as:

(NOTE: I have written it as (n,k) because I'm not sure how to format it properly using Latex like it is seen in the textbook, with the n above the k. If anyone knows how to do this please let me know).

Now, I am asked to prove that .

Let , then .

Also,

Adding them together gives: .

Now,

So to prove that I need to show:

At this point I know I have done something wrong, because I tried it out with real values (say n=6 and k=3), and they don't match:

Can anyone help?

Thanks in advance