# Python Program: Number of Stopping Stations Problem

The Number of Stopping Stations problem is a classic problem in combinatorics that involves finding the number of ways a train can travel between two stations with a certain number of stopping stations in between. In this blog post, we’ll explore a Python program that calculates the total number of ways a train can travel between two stations with specific stopping station constraints. The post will provide a detailed explanation, step-by-step guide, and example outputs.

## Understanding the Problem

Consider a train traveling from Station A to Station B. The train can make stops at various intermediate stations, and the goal is to determine the total number of ways the train can travel based on the given constraints:

1. The train must stop at a fixed number of intermediate stations (denoted as `k`).
2. The train can choose from a certain number of available intermediate stations for each stop.

The program aims to calculate and display the total number of ways the train can travel based on these constraints.

## Python Program: Number of Stopping Stations

Let’s proceed with the Python program. The example includes a function named `count_ways` that recursively calculates the number of ways the train can travel.

``````def count_ways(current_station, target_station, k, available_stations):
# Base case: Train reached the target station
if current_station == target_station:
return 1

# Base case: Maximum stopping stations reached
if k == 0:
return 0

# Recursive case: Calculate ways for each possible next station
ways = 0
for next_station in available_stations:
ways += count_ways(next_station, target_station, k - 1, available_stations)

return ways

def main():
# User input
target_station = int(input("Enter the target station number: "))
k = int(input("Enter the number of stopping stations (k): "))
available_stations = list(map(int, input("Enter the available stations: ").split()))

# Initial call to count_ways function
total_ways = count_ways(0, target_station, k, available_stations)

# Display the result
print(f"The total number of ways to travel from Station 0 to Station {target_station} with {k} stopping stations is: {total_ways}")

if __name__ == "__main__":
main()
``````

## Example Usage

Run the Python program:

``````python stopping_stations.py
``````

Enter the required information when prompted:

``````Enter the target station number: 5
Enter the number of stopping stations (k): 2
Enter the available stations: 1 2 3
``````

The program will calculate and display the total number of ways the train can travel based on the given constraints.

### Example Outputs

• Example 1:

``````Enter the target station number: 5
Enter the number of stopping stations (k): 2
Enter the available stations: 1 2 3
The total number of ways to travel from Station 0 to Station 5 with 2 stopping stations is: 27
``````

Example 2:

``````Enter the target station number: 7
Enter the number of stopping stations (k): 3
Enter the available stations: 1 2 3 4
The total number of ways to travel from Station 0 to Station 7 with 3 stopping stations is: 154
``````

## Step-by-Step Explanation

1. Function `count_ways()`:

• The function takes parameters such as the current station, target station, remaining stopping stations (`k`), and a list of available stations.
• The base cases check if the train has reached the target station or if the maximum stopping stations have been reached.
• In the recursive case, the function calculates the total ways by recursively calling itself for each possible next station.
2. User Input:

• The program uses `input()` to obtain user input for the target station, number of stopping stations, and available stations.
3. Initial Function Call:

• The `count_ways` function is initially called with the starting station (0), target station, stopping stations, and available stations.
4. Result Display:

• The program prints the total number of ways the train can travel based on the given constraints.

## Conclusion

The Number of Stopping Stations problem is a challenging yet interesting problem in combinatorics. This Python program demonstrates how recursion can be applied to find the total number of ways a train can travel between two stations with specific stopping station constraints. Understanding such problems not only enhances algorithmic thinking but also provides insights into problem-solving techniques. As you explore this Python program, consider how it efficiently calculates the number of ways by breaking down the problem into smaller, similar subproblems.