(b) That's right. C may have a good reason for wanting to enforce the contract that A entered into with B. After all, she lives in the same house as B and would benefit if the contract were performed. But C was not a party to the agreement, so A only owes contractual obligations to B. Under the doctrine of privity, C is a stranger to the contract (a 'third party') and has no right to bring an action to enforce it, even if the person who has that right chooses not to do so.
There are some important exceptions to the privity doctrine. If a contract has the effect of creating an 'equitable interest' for a third party, then that third party will be allowed to enforce the contract. An example of this is an insurance contract, where a third party is the beneficiary of the contract. Despite not being a party to the contract, the third party can bring an action to enforce it and obtain their equitable interest.