Binomial question

Q:
http://prntscr.com/s0lngg

My w/o:
http://prntscr.com/s0lnpd

Ans is n=2k+1 in the textbook. Where did I go wrong??

In your last line before you say "let y = n - k", you cancel a lot of the terms. But I think you are left with too much. On the RHS, where your last term before cancelling is (n - k), note that the term immediately before this will be (n - k + 1).

I’m confused... what term are you talking about? Are you saying I can turn k! into n-k+1?

No - I'm talking about the bit where you go from

(k + 1)! [n(n - 1)...(n - k + 1)] = k! [n(n - 1)...(n - k)],

which I agree with, to

(k + 1)! (n - k + 1) = k! (n - k),

which I don't. What I was suggesting was that you include one more term in the bit in the square bracket on the RHS so that it reads

(k + 1)! [n(n - 1)...(n - k + 1)] = k! [n(n - 1)...(n - k + 1)(n - k)],

as this might make it easier to see how this cancels down.

If I include n-k+1 on the RHS then it cancels down to (k+1)!=k!(n-k) => n = (k+1)!/k! + k
which gives me n = 2k + 1! Thanks 👍