To find the total number of paths from S to T without retracing steps, we can analyze the distinct segments of the journey. From S to U, there are 2 distinct paths. From S to V, there are 3 distinct paths. To reach T from U, there are 3 paths, and to reach T from V, there are 2 paths. The total number of routes is calculated by multiplying the number of ways to reach an intermediate point by the number of ways to proceed from that point to the destination, considering all possible intermediate points without repetition.