Question

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

• A. 92
• B. 90
• C. 91
• D. 93

Answer 1

The Correct Option is B

To solve this problem, we need to determine the number of different journey tickets required for a railway line with 10 stations.

Let's understand the situation clearly: 

• Each station can be denoted as a point in a line. If there are 10 stations, we can label them as A₁, A₂, ..., A₁₀.
• We are essentially looking for the number of different starting and ending points available for a journey ticket.

To calculate this, consider:

• Choosing a starting station can happen in 10 different ways.
• Choosing an ending station must be different from the starting station.

So for each starting station, there are (10 - 1) = 9 possible ending stations. Thus, for 10 stations:

\[\text{Total number of journey tickets} = 10 \times 9 = 90\]

This calculation shows that 90 different journey tickets are required by the authorities.

Therefore, the correct answer is 90.

Answer 2

If there are 10 stations, the number of ways to choose two distinct stations (i.e., the number of possible journeys between two stations) is given by the combination formula \( \binom{n}{2} \), where \( n \) is the total number of stations.

Thus, the number of different journey tickets is:

\[ \binom{10}{2} = \frac{10 \times 9}{2} = 45 \]

This is the number of possible journeys in one direction. Since a ticket can be for a journey in either direction (forward or backward), we multiply by 2:

\[ 45 \times 2 = 90 \]

Thus, the number of different journey tickets required is 90.