A, B and C are three places such that there are three different roads from A to B, four different roads from B to C and three different roads from A to C. In how many different ways can one travel from A to C using these roads?

The correct option is 15.

To find the total number of different ways to travel from place A to place C, we can analyze the available routes based on the paths described. We have two main cases for traveling from A to C:

Case 1: Traveling directly from A to C
The problem states that there are 3 different direct roads from A to C.
Therefore, the number of ways for Case 1 is 3.

Case 2: Traveling from A to C via B (A → B → C)
To travel from A to C through B, we must first travel from A to B, and then from B to C.
1. There are 3 different roads from A to B.
2. There are 4 different roads from B to C.
Using the fundamental counting principle (multiplication rule), the number of ways to perform these consecutive actions is:
$$3\times 4=12$$ Therefore, there are 12 ways for Case 2.

Total Number of Ways:
Since these two cases (direct route and route via B) are mutually exclusive, we use the addition rule to find the total number of ways:
$$\text{Total ways}=\text{Ways (Direct)}+\text{Ways (via B)}$$ $$\text{Total ways}=3+12=15$$ Thus, there are 15 different ways to travel from A to C.