There are 10 stations on a railway line. The number of different journey tickets that are required by the authorities, is

Options

A. 92
B. 90
C. 91
D. 93

Answer

To determine the number of different journey tickets required by the authorities for a railway line with 10 stations, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Stations**:
We have a total of 10 stations labeled as Station 1, Station 2, Station 3, ..., Station 10.

2. **Understanding the Journey Tickets**:
A journey ticket is required for traveling between two different stations. Therefore, we need to find the number of unique pairs of stations.

3. **Calculating the Number of Unique Pairs**:
To find the number of unique pairs of stations, we can use the combination formula:
\[ \text{Number of ways to choose 2 stations from 10} = \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
where \( n \) is the total number of stations (10) and \( r \) is the number of stations to choose (2).
\[ \binom{10}{2} = \frac{10!}{2!(10-2)!} = \frac{10 \times 9}{2 \times 1} = 45 \]

4. **Considering Directionality**:
Each journey can be made in two directions (from Station A to Station B and from Station B to Station A). Therefore, we need to multiply the number of unique pairs by 2:
\[ \text{Total journey tickets} = 45 \times 2 = 90 \]

5. **Conclusion**:
The total number of different journey tickets required by the authorities is 90.

### Final Answer:
The number of different journey tickets required is **90**.

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