Program Code

def generate_parentheses(n):
    """
    Generate all valid combinations of n pairs of parentheses.

    Parameters:
    n (int): The number of pairs of parentheses.

    Returns:
    List[str]: A list containing all valid combinations of n pairs of parentheses.
    """
    def backtrack(current, open_count, close_count):
        # If the current combination is of the maximum length
        if len(current) == 2 * n:
            result.append(current)
            return
        
        # Add an open parenthesis if we still have open parentheses left to add
        if open_count < n:
            backtrack(current + '(', open_count + 1, close_count)
        
        # Add a close parenthesis if we have open parentheses that can be closed
        if close_count < open_count:
            backtrack(current + ')', open_count, close_count + 1)

    result = []
    backtrack('', 0, 0)
    return result

# Example usage:
if __name__ == "__main__":
    n = 3  # You can change this value to generate different numbers of pairs
    combinations = generate_parentheses(n)
    print("Valid combinations of", n, "pairs of parentheses:", combinations)

Program Structure Explanation

This Python program consists of a main function generate_parentheses(n) and a nested helper function backtrack(current, open_count, close_count). Here’s how the program works:

• Function Definition:
  • The generate_parentheses function takes an integer n as input, representing the number of pairs of parentheses.
  • It initializes an empty list result to store valid combinations.
• Backtracking Algorithm:
  • The backtrack function is a recursive function responsible for building the combinations.
  • It takes the current string of parentheses, the count of open parentheses, and the count of close parentheses as parameters.
  • If the current combination reaches a length of 2 * n, it means we have a valid combination, which is added to result.
  • The function explores two main choices:
    • Add an open parenthesis if we still have open parentheses left to add (open_count < n).
    • Add a close parenthesis if we have an open parenthesis that can be closed (close_count < open_count).
• Execution:
  • The if __name__ == "__main__" block allows for testing the function by specifying a value for n and printing the resulting combinations.

How to Run the Program

To run this program, copy the code into a Python environment or a script file and execute it. Modify the value of n to generate different numbers of parentheses pairs. The result will be printed to the console.