There are four bus routes between A and B, and three 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. 144
C. 14
D. 19

Answer

To solve the problem, we need to determine the total number of ways a man can travel from A to C via B and then return back to A without using the same bus route more than once.

### Step-by-Step Solution:

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

2. **Calculate the Outbound Trip (A to C)**:
- When traveling from A to B, the man has 4 options (routes).
- After reaching B, he can choose any of the 3 routes to travel to C.
- Therefore, the total number of ways to travel from A to C is:
\[ \text{Ways from A to B} \times \text{Ways from B to C} = 4 \times 3 = 12 \text{ ways} \]

3. **Calculate the Return Trip (C to A)**:
- On the return trip from C to B, the man must not use the route he took from B to C. Since he had 3 options to go from B to C, he will have 2 remaining options to return from C to B.
- After reaching B, he must not use the route he took from A to B. Since he had 4 options to go from A to B, he will have 3 remaining options to return from B to A.
- Therefore, the total number of ways to return from C to A is:
\[ \text{Ways from C to B} \times \text{Ways from B to A} = 2 \times 3 = 6 \text{ ways} \]

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

### 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**.

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