Framed Slotted Aloha (FSA) protocols have been widely used in diverse multiple access communication systems. Consider an FSA system that consists of pre-specified users and there is assignable number of slots accommodated in a frame. Each user will randomly transmit a packet in one of the assigned slots. Characterizing the probability for the event that an exact number of collided slots occurs in a frame, is an important issue for some FSA protocols based networks, e.g., RFID networks. In this paper, we propose a generic analytical approach (GAA) to evaluate the above characterized probability of FSA systems by using the principle of inclusion and exclusion. Accuracy of the ECFF is validated via simulations. We believe that major contributions of this paper can provide valuable method for the evaluation of the characterized probability in a variety of FSA protocols based applications.