To find the number of distinct paths from S to T without retracing steps through U or V, we need to consider the possible segments of the journey. From S, one can go to U or V. If going via U, there are two distinct segments from U to T. If going via V, there are also two distinct segments from V to T. Since the path can involve going from S to U then to T, or from S to V then to T, the total number of routes is the sum of routes through each intermediate point. Additionally, a path could go from S to U, then to V, and then to T, or from S to V, then to U, and then to T. By applying the multiplication principle for independent choices at each stage and summing the possibilities for distinct routes, the total combinations can be calculated.