Break the number into three parts

Given a really large number, break it into 3 whole numbers such that they sum up to the original number and count number of ways to do so.

Examples :

Input : 3
Output : 10
The possible combinations where the sum
of the numbers is equal to 3 are:
0+0+3 = 3
0+3+0 = 3
3+0+0 = 3
0+1+2 = 3
0+2+1 = 3
1+0+2 = 3
1+2+0 = 3
2+0+1 = 3
2+1+0 = 3
1+1+1 = 3

Input : 6
Output : 28

A total of 10 ways, so answer is 10.



Naive Approach: Try all combinations from 0 to the given number and check if they add upto the given number or not, if they do, increase the count by 1 and continue the process.

C/C++

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// C++ program to count number of ways to break
// a number in three parts.
#include <bits/stdc++.h>
#define ll long long int
using namespace std;
  
// Function to count number of ways
// to make the given number n
ll count_of_ways(ll n)
{
    ll count = 0;
    for (int i = 0; i <= n; i++)
        for (int j = 0; j <= n; j++)
            for (int k = 0; k <= n; k++)
                if (i + j + k == n)
                    count++;
    return count;
}
  
// Driver Function
int main()
{
    ll n = 3;
    cout << count_of_ways(n) << endl;
    return 0;
}

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Java

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// Java program to count number of ways to break
// a number in three parts
import java.io.*;
  
class GFG {
    // Function to count number of ways
    // to make the given number n
    static long count_of_ways(long n)
    {
        long count = 0;
        for (int i = 0; i <= n; i++)
            for (int j = 0; j <= n; j++)
                for (int k = 0; k <= n; k++)
                    if (i + j + k == n)
                        count++;
        return count;
    }
  
    // driver program
    public static void main(String[] args)
    {
        long n = 3;
        System.out.println(count_of_ways(n));
    }
}
  
// Contributed by Pramod Kumar

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Python3

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# Python3 program to count number of 
# ways to break
# a number in three parts.
  
# Function to count number of ways
# to make the given number n
def count_of_ways(n):
  
    count = 0
    for i in range(0, n+1):
        for j in range(0, n+1):
            for k in range(0, n+1):
                if(i + j + k == n):
                    count = count + 1
    return count
  
# Driver Function
if __name__=='__main__':
    n = 3
    print(count_of_ways(n))
  
  
# This code is contributed by
# Sanjit_Prasad

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C#

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// C# program to count number of ways 
// to break a number in three parts
using System;
  
class GFG {
      
    // Function to count number of ways
    // to make the given number n
    static long count_of_ways(long n)
    {
        long count = 0;
        for (int i = 0; i <= n; i++)
            for (int j = 0; j <= n; j++)
                for (int k = 0; k <= n; k++)
                    if (i + j + k == n)
                        count++;
        return count;
    }
  
    // driver program
    public static void Main()
    {
        long n = 3;
        Console.WriteLine(count_of_ways(n));
    }
}
  
// This code is Contributed by vt_m.

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PHP

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<?php
// PHP program to count number 
// of ways to break a number 
// in three parts.
  
// Function to count number of ways
// to make the given number n
function count_of_ways( $n)
{
    $count = 0;
    for ($i = 0; $i <= $n; $i++)
        for ($j = 0; $j <= $n; $j++)
            for ($k = 0; $k <= $n; $k++)
                if ($i + $j + $k == $n)
                    $count++;
    return $count;
}
  
// Driver Code
$n = 3;
echo count_of_ways($n);
  
// This code is Contributed by vt_m.
?>

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Output :

10

Time Complexity : O(n3)

Efficient Approach: If we carefully observe the test cases then we realize that the number of ways to break a number n into 3 parts is equal to (n+1) * (n+2) / 2.

C/C++

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// C++ program to count number of ways to break
// a number in three parts.
#include <bits/stdc++.h>
#define ll long long int
using namespace std;
  
// Function to count number of ways
// to make the given number n
ll count_of_ways(ll n)
{
    ll count;
    count = (n + 1) * (n + 2) / 2;
    return count;
}
  
// Driver Function
int main()
{
    ll n = 3;
    cout << count_of_ways(n) << endl;
    return 0;
}

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Java

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// Java program to count number of ways to break
// a number in three parts
import java.io.*;
  
class GFG {
    // Function to count number of ways
    // to make the given number n
    static long count_of_ways(long n)
    {
        long count = 0;
        count = (n + 1) * (n + 2) / 2;
        return count;
    }
  
    // driver program
    public static void main(String[] args)
    {
        long n = 3;
        System.out.println(count_of_ways(n));
    }
}
  
// Contributed by Pramod Kumar

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Python3

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# Python 3 program to count number of 
# ways to break a number in three parts.
  
# Function to count number of ways
# to make the given number n
def count_of_ways(n):
    count = 0
    count = (n + 1) * (n + 2) // 2
    return count
  
# Driver code
n = 3
print(count_of_ways(n))
  
# This code is contributed by Shrikant13

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C#

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// C# program to count number of ways to
// break a number in three parts
using System;
  
class GFG {
      
    // Function to count number of ways
    // to make the given number n
    static long count_of_ways(long n)
    {
        long count = 0;
        count = (n + 1) * (n + 2) / 2;
        return count;
    }
  
    // driver program
    public static void Main()
    {
        long n = 3;
        Console.WriteLine(count_of_ways(n));
    }
}
  
// This code is Contributed by vt_m.

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PHP

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<?php
// PHP program to count number 
// of ways to break a number
// in three parts.
  
// Function to count number of ways
// to make the given number n
function count_of_ways( $n)
{
    $count;
    $count = ($n + 1) * ($n + 2) / 2;
    return $count;
}
  
// Driver Code
$n = 3;
echo count_of_ways($n);
  
// This code is Contributed by vt_m.
?>

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Output :

10

Time Complexity: O(1)

This article is contributed by Aditya Gupta. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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