In long term evolution-advanced (LTE-A) networks, the carrier aggregation technique is incorporated for user equipments (UEs) to simultaneously aggregate multiple component carriers (CCs) for achieving higher transmission rate. Many research works for LTE-A systems with carrier aggregation configuration have concentrated on the radio resource management problem for downlink transmission, including mainly CC assignment and packet scheduling. Most previous studies have not considered that the assigned CCs in each UE can be changed. Furthermore, they also have not considered the modulation and coding scheme constraint, as specified in LTE-A standards. Therefore, their proposed schemes may limit the radio resource usage and are not compatible with LTE-A systems. In this paper, we assume that the scheduler can reassign CCs to each UE at each transmission time interval and formulate the downlink radio resource scheduling problem under the modulation and coding scheme constraint, which is proved to be NP-hard. Then, a novel greedy-based scheme is proposed to maximize the system throughput while maintaining proportional fairness of radio resource allocation among all UEs. We show that this scheme can guarantee at least half of the performance of the optimal solution. Simulation results show that our proposed scheme outperforms the schemes in previous studies.