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

Options

A. a. 90
B. b. 45
C. c. 135
D. d. 100

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 1: Understand the Problem
We have 10 stations, and we need to find the number of different tickets that can be issued for journeys between these stations. A ticket is required for each unique journey from one station to another.

### Step 2: Determine the Combinations
A journey ticket can be represented by choosing 2 stations out of the 10. The order of selection does not matter (i.e., traveling from station A to station B is the same as traveling from station B to station A). Therefore, we use combinations to find the number of ways to choose 2 stations from 10.

### Step 3: Use the Combination Formula
The formula for combinations is given by:
\[ nCk = \frac{n!}{k!(n-k)!} \]
where \( n \) is the total number of items (stations), \( k \) is the number of items to choose, and \( ! \) denotes factorial.

In our case, \( n = 10 \) and \( k = 2 \):
\[ 10C2 = \frac{10!}{2!(10-2)!} = \frac{10!}{2! \cdot 8!} \]

### Step 4: Simplify the Calculation
Now, we can simplify \( 10C2 \):
\[ 10C2 = \frac{10 \times 9 \times 8!}{2! \times 8!} \]
The \( 8! \) cancels out:
\[ 10C2 = \frac{10 \times 9}{2!} = \frac{10 \times 9}{2 \times 1} = \frac{90}{2} = 45 \]

### Step 5: Total Journey Tickets
Since a journey can be made in either direction (from station A to station B or from station B to station A), the total number of different journey tickets is:
\[ \text{Total Tickets} = 45 \text{ (from A to B)} + 45 \text{ (from B to A)} = 90 \]

### Final Answer
Thus, the number of different journey tickets required by the authorities is **90**.

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