Final answer:
The coordinates of the center of mass for the system, including the ring and beads, are determined by the symmetry of the arrangement. As the beads are equidistant from the center of the ring and have the same mass, the overall center of mass coincides with the center of the ring, resulting in coordinates (0, 0).
Step-by-step explanation:
The center of mass of the system consisting of the ring and the beads can be found by considering the individual center of masses and the masses of the ring and the beads. Since all beads have the same mass and are equidistant from the center of the ring, the center of mass of the beads is at the center of the ring. Therefore, the coordinates of the center of mass of the system are the same as the coordinates of the center of the ring, which is (0, 0). The symmetry of the arrangement simplifies the calculation, demonstrating that the system's center of mass is at the geometric center of the ring.