There are five routes for a journey from stations . A to B . In how many different ways can a man go from A to B and return , if for returning.
the same route is not taken?

Answer

To solve the problem of how many different ways a man can go from station A to station B and return without taking the same route back, we can break it down into clear steps.

### Step-by-Step Solution:

1. **Identify the Number of Routes**:
There are 5 different routes available for the journey from station A to station B.

2. **Choose a Route to Go from A to B**:
The man can choose any one of the 5 routes to go from A to B. Therefore, there are 5 choices for the outbound journey.

3. **Choose a Different Route to Return from B to A**:
Since the man cannot take the same route back, he has to choose from the remaining routes. After choosing one route to go to B, there are 4 routes left to return to A.

4. **Calculate the Total Number of Ways**:
The total number of different ways to complete the journey (going to B and returning to A) can be calculated by multiplying the number of choices for the outbound journey by the number of choices for the return journey:
\[ \text{Total Ways} = (\text{Choices for A to B}) \times (\text{Choices for B to A}) = 5 \times 4 = 20 \]

Thus, the total number of different ways the man can go from A to B and return without taking the same route is **20**.