There are five stations on a railway line. What is the number of different journey tickets that are required for railway authorities?

Options

A. 25
B. 20
C. 30
D. 35

Answer

To solve the problem of how many different journey tickets are required for five stations on a railway line, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Stations**:
We have 5 stations, which we can label as A, B, C, D, and E.

2. **Understanding the Ticket Requirement**:
A journey ticket is required for traveling from one station to another. Therefore, we need to consider all possible pairs of stations.

3. **Determine the Number of Tickets**:
For each journey, we need to select 2 stations from the 5 available stations. The order in which we select the stations matters (i.e., traveling from A to B is different from traveling from B to A).

4. **Use the Permutation Formula**:
The number of ways to choose and arrange 2 stations from 5 can be calculated using the permutation formula:
\[ P(N, R) = \frac{N!}{(N-R)!} \]
where \(N\) is the total number of stations (5) and \(R\) is the number of stations to choose (2).

5. **Substituting Values**:
Here, \(N = 5\) and \(R = 2\):
\[ P(5, 2) = \frac{5!}{(5-2)!} = \frac{5!}{3!} \]

6. **Calculating Factorials**:
We know that:
\[ 5! = 5 \times 4 \times 3! \]
Therefore, substituting this into our equation gives:
\[ P(5, 2) = \frac{5 \times 4 \times 3!}{3!} \]
The \(3!\) cancels out:
\[ P(5, 2) = 5 \times 4 = 20 \]

7. **Conclusion**:
Thus, the total number of different journey tickets required for the railway authorities is **20**.