To generate a valid combination with the parenthesis is a try and error process and we will solve this problem with the recursion approach.

To solve this problem, we will follow these steps,

1. We have to initialize a count variable to zero, and an open variable is for open parenthesis, and a close variable is for close parenthesis.
2. Initialize a vector to store the strings.
3. Whenever the open is less than the number then we add an open parenthesis.
4. Whenever the open parenthesis is grater the closing parenthesis then we add closing parenthesis to it.
5. If the value of open parenthesis equals the number then we add the strings into the vector.

C++ Implementation:

#include <bits/stdc++.h>
using namespace std;

void traverse(int open, int close, int n, int& count, string str, vector<string>& v)
{
    if (close == n) {
        v.push_back(str);
        return;
    }
    if (open < n) {
        traverse(open + 1, close, n, count, str + "(", v);
    }
    if (close < open) {
        traverse(open, close + 1, n, count, str + ")", v);
    }
    return;
}

void genarate_parenthesis(int num)
{
    string str = "";
    int open_brace = 0, close_brace = 0, count = 0;
    vector<string> v;
    traverse(open_brace, close_brace, num, count, str, v);
    cout << "Posiible combination : ";
    for (int i = 0; i < v.size(); i++) {
        cout << v[i] << endl;
    }
}

int main()
{
    int t;

    cout << "TestCase : ";
    cin >> t;

    while (t--) {
        int num;

        cout << "Enter the number: ";
        cin >> num;

        genarate_parenthesis(num);
    }
    
    return 0;
}

Output

TestCase : 3
Enter the number: 4
(((())))
((()()))
((())())
((()))()
(()(()))
(()()())
(()())()
(())(())
(())()()
()((()))
()(()())
()(())()
()()(())
()()()()
Enter the number: 3
((()))
(()())
(())()
()(())
()()()
Enter the number: 5
((((()))))
(((()())))
(((())()))
(((()))())
(((())))()
((()(())))
((()()()))
((()())())
((()()))()
((())(()))
((())()())
((())())()
((()))(())
((()))()()
(()((())))
(()(()()))
(()(())())
(()(()))()
(()()(()))
(()()()())
(()()())()
(()())(())
(()())()()
(())((()))
(())(()())
(())(())()
(())()(())
(())()()()
()(((())))
()((()()))
()((())())
()((()))()
()(()(()))
()(()()())
()(()())()
()(())(())
()(())()()
()()((()))
()()(()())
()()(())()
()()()(())
()()()()()