Break the number into three parts - GeeksforGeeks

06 May, 2021
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++
Java
Python3
C#
PHP
Javascript
// C++ program to count number of ways to break// a number in three parts.#include <bits/stdc++.h>#define ll long long intusing namespace std; // Function to count number of ways// to make the given number nll 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 Functionint main(){    ll n = 3;    cout << count_of_ways(n) << endl;    return 0;}
// Java program to count number of ways to break// a number in three partsimport 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
# 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 ndef 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 Functionif __name__=='__main__':    n = 3    print(count_of_ways(n))  # This code is contributed by# Sanjit_Prasad
// C# program to count number of ways// to break a number in three partsusing 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.
<?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 nfunction 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.?>
<script> // JavaScript program to count// number of ways to break// a number in three parts. // Function to count number of ways// to make the given number nfunction count_of_ways(n){    let count = 0;    for(let i = 0; i <= n; i++)        for(let j = 0; j <= n; j++)            for(let k = 0; k <= n; k++)                if (i + j + k == n)                    count++;                         return count;} // Driver codelet n = 3; document.write(count_of_ways(n) + "<br>"); // This code is contributed by Surbhi Tyagi. </script>
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++
Java
Python3
C#
PHP
Javascript
// C++ program to count number of ways to break// a number in three parts.#include <bits/stdc++.h>#define ll long long intusing namespace std; // Function to count number of ways// to make the given number nll count_of_ways(ll n){    ll count;    count = (n + 1) * (n + 2) / 2;    return count;} // Driver Functionint main(){    ll n = 3;    cout << count_of_ways(n) << endl;    return 0;}
// Java program to count number of ways to break// a number in three partsimport 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
# 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 ndef count_of_ways(n):    count = 0    count = (n + 1) * (n + 2) // 2    return count # Driver coden = 3print(count_of_ways(n)) # This code is contributed by Shrikant13
// C# program to count number of ways to// break a number in three partsusing 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.
<?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 nfunction 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.?>
<script> // javascript 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)    {        var count = 0;        count = (n + 1) * (n + 2) / 2;        return count;    }      // driver program         var n = 3;        document.write(count_of_ways(n)); // This code is contributed by bunnyram19.</script>
Output : 
10
Time Complexity: O(1)
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vt_m
Sanjit_Prasad
shrikanth13
surbhityagi15
bunnyram19
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