A boy lives at X and wants to go to School at Z. From his home X he has to first reach Y and then Y to Z. He may go X to Y by either 3 bus routes or 2 train routes. From there, he can either choose 4 bus routes or 5 train routes to reach Z. How many ways are there to go from X to Z?

Step 1 (Calculate routes from X to Y): The boy can travel from home $X$ to intermediate station $Y$ by either 3 bus routes or 2 train routes. The total ways to go from $X$ to $Y$ is:

Step 2 (Calculate routes from Y to Z): The boy can travel from $Y$ to school $Z$ by either 4 bus routes or 5 train routes. The total ways to go from $Y$ to $Z$ is:

Step 3 (Multiply using product rule): Since the trip from $X$ to $Z$ requires sequential choices (first $X\to Y$, then $Y\to Z$), we apply the product rule of counting:

Step 4 (Conclusion): There are exactly 45 distinct ways to travel from X to Z.