Regular grammar with at most one c I am attempting to make a regular grammar over alphabet {a, b, c} where there is at most one c. So far, I have the regular expression ((a|b)*|c)(a|b)* but am unsure where to go from here; my previous attempts have ended up allowing multiple c's.

The solution I has gives (N={s,t}, T={a,b,c}, s, R), s→є, s→as, s→bs, s→ct, t→at, t→bt, t→є, however I do not see how this limits the number of c's generated to at most 1.
The rule s->ct can only be done once, becuase after that there is no more s left, and from t you cannot generate more s. Thus at most one c can be generated, because no other rule generates c.  

Also the rest of the grammar looks fine.