Eight railway stations A, B, C, D, E, F, G and H are connected either by two-way passages or one-way passages. One-way passages are from C to A, E to G, B to F, D to H, G to C, E to C and H to G. Two-way passages are between A and E, G and B, F and D, and E and D.
In how many different ways can a train travel from F to A without passing through any station more than once?

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:

1. From F, the only initial movement is F ↔ D.
2. 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.