# 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++

`// 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

`// 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

`# 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#

`// 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

`<?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(n^{3})

**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++

`// 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

`// 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

`# 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#

`// 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

`<?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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