Direction for the following 3 (three) items: Consider the given information and answer the three items (76-78) that follow.

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?

The correct answer is Option 4 (4).

Step-by-step Explanation:

Let us first analyze the connections between the 8 railway stations A, B, C, D, E, F, G, and H as given in the problem statement:

One-way passages:
• C → A
• E → G
• B → F
• D → H
• G → C
• E → C
• H → G
• F → D (Wait, let us list two-way passages carefully)

Two-way passages (can travel in either direction):
• A ↔ E
• G ↔ B
• F ↔ D
• E ↔ D

We are asked to find the total number of different valid paths from station F to station A without visiting any station more than once.

Let us trace all possible paths starting from station F:

1. From station F, the available connections are:
- A two-way passage to D (F ↔ D). Note that B → F is a one-way passage into F, so we cannot go from F to B.

So, every valid path from F must start with: F → D.

2. From station D, we can move to:
• Station H (via D → H)
• Station E (via D ↔ E)

Let us split into two main branches from station D:

Branch 1: F → D → H
From H, the only outgoing passage is H → G.
So the path continues: F → D → H → G.
From G, the outgoing passages are G → C and G ↔ B.
- Sub-path 1a: Go to C (via G → C).
From C, the only outgoing passage is C → A.
This gives the valid complete path: F → D → H → G → C → A.
- Sub-path 1b: Go to B (via G ↔ B).
From B, the only outgoing passage is B → F. However, station F has already been visited, so this path cannot be extended further without visiting a station twice.

Branch 2: F → D → E
From E, the available outgoing options are:
• Direct passage to A (via E ↔ A):
This gives the valid complete path: F → D → E → A.
• Passage to C (via E → C):
From C, the outgoing passage is C → A.
This gives the valid complete path: F → D → E → C → A.
• Passage to G (via E → G):
From G, the available passage leading towards A is G → C.
From C, we go C → A.
This gives the valid complete path: F → D → E → G → C → A.

Summary of all valid paths from F to A:
1. F → D → E → A
2. F → D → E → C → A
3. F → D → E → G → C → A
4. F → D → H → G → C → A

Thus, there are exactly 4 different ways for a train to travel from station F to station A without passing through any station more than once.