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Count of ways to write N as a sum of three numbers
  • Last Updated : 20 Apr, 2021

Given a positive integer N, count number of ways to write N as a sum of three numbers. For numbers which are not expressible print -1.
Examples:
 

Input: N = 4 
Output:
Explanation: 
( 1 + 1 + 2 ) = 4 
( 1 + 2 + 1 ) = 4 
( 2 + 1 + 1 ) = 4. 
So in total, there are 3 ways.
Input: N = 5 
Output:
( 1 + 1 + 3 ) = 5 
( 1 + 3 + 1 ) = 5 
( 3 + 1 + 1 ) = 5 
( 1 + 2 + 2 ) = 5 
( 2 + 2 + 1 ) = 5 
( 2 + 1 + 2 ) = 5. 
So in total, there are 6 ways 
 

 

Approach: To solve the problem mentioned above if we take a closer look we will observe a pattern in solution to the question. For all the numbers that are greater than 2 we get a series 3, 6, 10, 15, 25 and so on, which is nothing but the sum of first N-1 natural numbers.
Below is the implementation of the above approach: 
 

C++




// C++ program to count the total number of
// ways to write N as a sum of three numbers
 
#include <bits/stdc++.h>
using namespace std;
 
// Function to find the number of ways
void countWays(int n)
{
    // Check if number is less than 2
    if (n <= 2)
        cout << "-1";
 
    else {
        // Calculate the sum
        int ans = (n - 1) * (n - 2) / 2;
 
        cout << ans;
    }
}
 
// Driver code
int main()
{
    int N = 5;
 
    countWays(N);
 
    return 0;
}

Java




// Java program to count the total number of
// ways to write N as a sum of three numbers
class GFG{
 
// Function to find the number of ways
static void countWays(int n)
{
     
    // Check if number is less than 2
    if (n <= 2)
    {
        System.out.print("-1");
    }
    else
    {
         
        // Calculate the sum
        int ans = (n - 1) * (n - 2) / 2;
        System.out.print(ans);
    }
}
 
// Driver code
public static void main(String[] args)
{
    int N = 5;
    countWays(N);
}
}
 
// This code is contributed by Amit Katiyar

Python3




# Python3 program to count the total number of
# ways to write N as a sum of three numbers
def countWays(N):
 
    # Check if number is less than 2
    if (N <= 2):
        print("-1")
    else:
         
        # Calculate the sum
        ans = (N - 1) * (N - 2) / 2
         
    print(ans)
 
# Driver code
if __name__ == '__main__':
     
    N = 5
    countWays(N)
 
# This code is contributed by coder001

C#




// C# program to count the total number of
// ways to write N as a sum of three numbers
using System;
 
class GFG{
     
// Function to find the number of ways
static void countWays(int n)
{
         
    // Check if number is less than 2
    if (n <= 2)
    {
        Console.WriteLine("-1");
    }
    else
    {
 
        // Calculate the sum
        int ans = (n - 1) * (n - 2) / 2;
        Console.WriteLine(ans);
    }
}
 
// Driver code    
static void Main()
{
    int N = 5;
    countWays(N);
}
}
 
// This code is contributed by divyeshrabadiya07   

Javascript




<script>
 
// Javascript program to count the total number of
// ways to write N as a sum of three numbers
 
// Function to find the number of ways
function countWays(n)
{
    // Check if number is less than 2
    if (n <= 2)
        document.write( "-1");
 
    else {
        // Calculate the sum
        var ans = (n - 1) * (n - 2) / 2;
 
        document.write( ans);
    }
}
 
// Driver code
 
var N = 5;
countWays(N);
 
</script>
Output: 
6

 

Time Complexity: O(1)
 

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