Your (a) looks good.
I'll use uppercase L since lowercase looks like 1.
For (b), the number k ranges from 0 to floor(n/3). When k = 0 there is only one way to order the letters. When k = 1 there are L ways to arrange the letters. When k = 2 there are C(L,2) ways to arrange the letters. Etc. So I get
which is the same as you got not counting the stray "=" sign.
Part (c) your letters get a bit mixed up. No need to call x the length of the polymer because that's already specified as p in ; and I would just use the letter k as before to reduce confusion. And I think you switched the minus sign to plus, I get
Part (d). I have another issue with this problem. Are they interested in going from to or to ? I'll assume the first.
Seems there should be a straightforward way, but since they don't place any restrictions on your recurrence, you could just write
Obviously it is justified by part (c). This satisfies what the question asks and in my opinion should be marked as correct. (Assuming I haven't made any mistakes leading up to it.)