Correct Option (4)
To determine the number of distinct ways a train can travel from station F to station A without traversing any station more than once, we must analyze the given one-way and two-way passages. The connections are as follows:
- One-way passages: C → A, E → G, B → F, D → H, G → C, E → C, H → G.
- Two-way passages: A ↔ E, G ↔ B, F ↔ D, E ↔ D.
Tracing all possible non-repeating paths from F to A:
- From F, the only initial movement is F ↔ D.
- From D, two main branches are possible: D ↔ E or D → H.
Considering these branches:
- Branch 1: F → D → E
- Path 1: F → D → E → A (using A ↔ E)
- Path 2: F → D → E → C → A (using E → C and C → A)
- Path 3: F → D → E → G → C → A (using E → G, G → C, and C → A)
- Branch 2: F → D → H
- Path 4: F → D → H → G → C → A (using D → H, H → G, G → C, and C → A)
All identified paths are distinct and do not involve revisiting any station. Therefore, there are 4 different ways a train can travel from F to A without passing through any station more than once.
Incorrect Options:
Options 1, 2, and 3 are incorrect because they do not represent the complete enumeration of all possible distinct paths from station F to station A, adhering to the condition of not passing through any station more than once. As demonstrated, there are exactly 4 such valid paths.