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This paper studies the stabilization problem for a class of networked control systems (NCSs) with communication constraints and random packet dropouts. The considered plant is equipped with multiple sensor nodes and actuator nodes. Only one of them can communicate with the controller at each transmission instant due to the communication constraints, and packet dropouts may occur at each transmission. A new transmission mechanism and state feedback controllers are considered, and the closed-loop NCS is modeled as a discrete-time switched system with a stochastic parameter. Sufficient conditions are derived for the closed-loop NCS to be exponentially mean square stable and a controller design procedure is also presented based on the obtained conditions. The effectiveness of the proposed method is shown by an illustrative example.