From IIT Roorkee to IIM Kashipur in Uttarakhand, there are total 325 railway stations falling on a straight railway line. A local train stops at every 3rd station, an express train stops at every 5th station and a bullet train stops at every 8th railway station. How many pairs of stations are connected by exactly two different types of trains?

Options

A. 25
B. 35
C. 36
D. 42

Answer

To solve the problem of how many pairs of stations are connected by exactly two different types of trains, we will follow these steps:

### Step 1: Identify the stopping patterns of the trains
- Local train stops at every 3rd station.
- Express train stops at every 5th station.
- Bullet train stops at every 8th station.

### Step 2: Find the least common multiples (LCM) of the stopping patterns
- **Local and Express Trains**: LCM of 3 and 5 is 15.
- **Express and Bullet Trains**: LCM of 5 and 8 is 40.
- **Bullet and Local Trains**: LCM of 8 and 3 is 24.

### Step 3: Determine how many stations each pair of trains stops at
- **Local and Express Trains**:
\[ \text{Number of stations} = \left\lfloor \frac{325}{15} \right\rfloor = 21 \]

- **Express and Bullet Trains**:
\[ \text{Number of stations} = \left\lfloor \frac{325}{40} \right\rfloor = 8 \]

- **Bullet and Local Trains**:
\[ \text{Number of stations} = \left\lfloor \frac{325}{24} \right\rfloor = 13 \]

### Step 4: Add the number of stations for each pair
Now, we add the number of stations where each pair of trains stops:
\[ 21 \text{ (Local and Express)} + 8 \text{ (Express and Bullet)} + 13 \text{ (Bullet and Local)} = 42 \]

### Conclusion
The total number of pairs of stations connected by exactly two different types of trains is **42**.