Binary manipulators have actuators with only two stable states. Therefore, they can reach only a finite number of locations. Compared to a manipulator built with continuous actuators, a binary manipulator provides reasonable performance, robustness, and is relatively inexpensive (up to an order of magnitude cheaper). The number of states attainable by a binary manipulator grows exponentially with the number of actuators. This makes brute force calculation of the inverse kinematics impossible for binary manipulators with many actuators. This paper presents a combinatorial method for planning trajectories for a binary manipulator while reducing the search space to a manageable size. Despite the discrete nature of binary actuation, it also creates motions that follow a specified trajectory accurately (in both position and orientation) without large deviations of the end-effector from the specified path.