There are 4 bus routes between A and B and 3 bus routes between B and C. A man can travel round trip in number of ways by bus from A to C via B. If he does not want to use a bus route more than once, in how many ways can he make round trip?

Options

A. 72
B. 142
C. 14
D. 19

Answer

To solve the problem of how many ways a man can make a round trip from A to C via B without using the same bus route more than once, we can break it down step by step.

### Step-by-Step Solution:

1. **Identify the Routes:**
- There are 4 bus routes from A to B.
- There are 3 bus routes from B to C.

2. **Calculate Ways from A to C:**
- The man can choose any of the 4 routes from A to B.
- For each of these 4 routes, he can choose any of the 3 routes from B to C.
- Therefore, the total number of ways to go from A to C is:
\[ \text{Ways from A to C} = 4 \times 3 = 12 \]

3. **Calculate Ways for the Return Trip (C to A):**
- On the return trip, he cannot use the bus routes he used on the way to C.
- Since he used one route from B to C, only 2 routes remain available to go back from C to B.
- He can still use any of the 3 routes from B to A (since he hasn't used them yet).
- Therefore, the total number of ways to return from C to A is:
\[ \text{Ways from C to A} = 2 \times 3 = 6 \]

4. **Calculate Total Round Trip Ways:**
- The total number of ways to complete the round trip from A to C and back to A is the product of the ways to go and the ways to return:
\[ \text{Total Round Trip Ways} = 12 \times 6 = 72 \]

### Final Answer:
The total number of ways the man can make a round trip from A to C via B without using the same bus route more than once is **72 ways**.