A novel synchronization scheme is proposed for a class of chaotic systems, extending the concept of multi-switching synchronization to combination synchronization such that the state variables of two or more driving systems synchronize with different state variables of the response system, simultaneously. The new scheme, multi-switching combination synchronization (MSCS), represents a significant extension of earlier multi-switching schemes in which two chaotic systems, in a driver-response configuration, are multi-switched to synchronize up to a scaling factor. In MSCS, the chaotic driving systems multi-switch a response chaotic system in combination synchronization. For certain choices of the scaling factors, MSCS reduces to multi-switching synchronization, implying that the latter is a special case of MSCS. A theoretical approach to control design, based on backstepping, is presented and validated using numerical simulations.