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Number of ways to insert two pairs of parentheses into a string of N characters
  • Last Updated : 01 Nov, 2019

Given a string str of length N, the task is to find the number of ways to insert only 2 pairs of parentheses into the given string such that the resultant string is still valid.

Examples:

Input: str = “ab”
Output: 6
((a))b, ((a)b), ((ab)), (a)(b), (a(b)), a((b))
which are a total of 6 ways.

Input: str = “aab”
Output: 20

Approach: it can be observed that for the lengths of the string 1, 2, 3, …, N a series will be formed as 1, 6, 20, 50, 105, 196, 336, 540, … whose Nth term is (N + 1)2 * ((N + 1)2 – 1) / 12.



Below is the implementation of the above approach:

C++

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// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
  
// Function to return the number of ways
// to insert the bracket pairs
int cntWays(string str, int n)
{
    int x = n + 1;
    int ways = x * x * (x * x - 1) / 12;
    return ways;
}
  
// Driver code
int main()
{
    string str = "ab";
    int n = str.length();
  
    cout << cntWays(str, n);
  
    return 0;
}

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Java

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// Java implementation of the approach
import java.util.*;
  
class GFG
{
  
// Function to return the number of ways
// to insert the bracket pairs
static int cntWays(String str, int n)
{
    int x = n + 1;
    int ways = x * x * (x * x - 1) / 12;
    return ways;
}
  
// Driver code
public static void main(String []args) 
{
    String str = "ab";
    int n = str.length();
  
    System.out.println(cntWays(str, n));
}
}
  
// This code is contributed by Rajput-Ji

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Python3

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# Python3 implementation of the approach 
  
# Function to return the number of ways 
# to insert the bracket pairs 
def cntWays(string, n) :
  
    x = n + 1
    ways = x * x * (x * x - 1) // 12
    return ways; 
  
# Driver code 
if __name__ == "__main__"
  
    string = "ab"
    n = len(string); 
  
    print(cntWays(string, n)); 
  
# This code is contributed by AnkitRai01

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C#

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// C# implementation of the approach
using System;
      
class GFG
{
  
// Function to return the number of ways
// to insert the bracket pairs
static int cntWays(String str, int n)
{
    int x = n + 1;
    int ways = x * x * (x * x - 1) / 12;
    return ways;
}
  
// Driver code
public static void Main(String []args) 
{
    String str = "ab";
    int n = str.Length;
  
    Console.WriteLine(cntWays(str, n));
}
}
  
// This code is contributed by Rajput-Ji

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Output:

6

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