Let break it into two parts one set X containing element which can be used only ones i.e 1.a, 1.b , another set Y contains all elements which have unlimited repeatation like b,c. r-combination means choose r elements from these two.
We have several choices like pick 1 element from set X and rest r-1 from Y, another case is choose 2 elements from X and rest (r-2) from Y and so on....upto choosing 0 elements from X and all from Y.
Choosing from X set is simple choosing j element from x elements, and r-j remaining from set Y is choosing like " (r+n-1)C(r) where we need r-combination from n element which have infinite copies".
So atlast we sum up all the cases :
$\sum_{j}^{j\leq x}\left ( \binom{x}{j} \binom{(r-j+y-1)}{r-j}\right )$
Please tell me if there is any mistake, I have just tried to do it.