A, B and C are three cities. There are 5 routes from A to B and 3 routes from B to C. How many different routes are there from A to C via B?

Answer

To solve the problem of finding the number of different routes from city A to city C via city B, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Routes from A to B**:
- There are 5 different routes from city A to city B.

2. **Identify the Routes from B to C**:
- There are 3 different routes from city B to city C.

3. **Calculate the Total Routes from A to C via B**:
- To find the total number of routes from A to C via B, we can use the multiplication principle of counting.
- For each route from A to B, there are multiple options to go from B to C. Specifically, for each of the 5 routes from A to B, there are 3 options to go from B to C.
- Therefore, the total number of different routes from A to C via B can be calculated as:
\[ \text{Total Routes} = (\text{Routes from A to B}) \times (\text{Routes from B to C}) = 5 \times 3 \]

4. **Perform the Calculation**:
- Now, we perform the multiplication:
\[ 5 \times 3 = 15 \]

5. **Conclusion**:
- Thus, the total number of different routes from city A to city C via city B is 15.

### Final Answer:
The total number of different routes from A to C via B is **15**.