# Ways to write n as sum of two or more positive integers

For a given number n > 0, find the number of different ways in which n can be written as a sum of at two or more positive integers.

Examples:

```Input : n = 5
Output : 6
Explanation : All possible six ways are :
4 + 1
3 + 2
3 + 1 + 1
2 + 2 + 1
2 + 1 + 1 + 1
1 + 1 + 1 + 1 + 1

Input : 4
Output : 4
Explanation : All possible four ways are :
3 + 1
2 + 2
2 + 1 + 1
1 + 1 + 1 + 1
```

## Recommended: Please solve it on “PRACTICE ” first, before moving on to the solution.

This problem can be solved in the similar fashion as coin change problem, the difference is only that in this case we should iterate for 1 to n-1 instead of particular values of coin as in coin-change problem.

## C/C++

 `// Program to find the number of ways, n can be ` `// written as sum of two or more positive integers. ` `#include ` `using` `namespace` `std; ` ` `  `// Returns number of ways to write n as sum of ` `// two or more positive integers ` `int` `countWays(``int` `n) ` `{ ` `    ``// table[i] will be storing the number of ` `    ``// solutions for value i. We need n+1 rows ` `    ``// as the table is consturcted in bottom up ` `    ``// manner using the base case (n = 0) ` `    ``int` `table[n+1]; ` ` `  `    ``// Initialize all table values as 0 ` `    ``memset``(table, 0, ``sizeof``(table)); ` ` `  `    ``// Base case (If given value is 0) ` `    ``table = 1; ` ` `  `    ``// Pick all integer one by one and update the ` `    ``// table[] values after the index greater ` `    ``// than or equal to n ` `    ``for` `(``int` `i=1; i

## Java

 `// Program to find the number of ways,  ` `// n can be written as sum of two or  ` `// more positive integers. ` `import` `java.util.Arrays; ` ` `  `class` `GFG { ` `     `  `    ``// Returns number of ways to write ` `    ``// n as sum of two or more positive  ` `    ``// integers ` `    ``static` `int` `countWays(``int` `n) ` `    ``{ ` `         `  `        ``// table[i] will be storing the  ` `        ``// number of solutions for value ` `        ``// i. We need n+1 rows as the  ` `        ``// table is consturcted in bottom ` `        ``// up manner using the base case ` `        ``// (n = 0) ` `        ``int` `table[] = ``new` `int``[n + ``1``]; ` `     `  `        ``// Initialize all table values as 0 ` `        ``Arrays.fill(table, ``0``); ` `     `  `        ``// Base case (If given value is 0) ` `        ``table[``0``] = ``1``; ` `     `  `        ``// Pick all integer one by one and ` `        ``// update the table[] values after  ` `        ``// the index greater than or equal  ` `        ``// to n ` `        ``for` `(``int` `i = ``1``; i < n; i++) ` `            ``for` `(``int` `j = i; j <= n; j++) ` `                ``table[j] += table[j - i]; ` `     `  `        ``return` `table[n]; ` `    ``} ` `     `  `    ``//driver code ` `    ``public` `static` `void` `main (String[] args) ` `    ``{ ` `        ``int` `n = ``7``; ` `         `  `        ``System.out.print(countWays(n)); ` `    ``} ` `} ` ` `  `// This code is contributed by Anant Agarwal. `

## Python

 `# Program to find the number of ways, n can be ` `# written as sum of two or more positive integers. ` ` `  `# Returns number of ways to write n as sum of ` `# two or more positive integers ` `def` `CountWays(n): ` ` `  `    ``# table[i] will be storing the number of ` `    ``# solutions for value i. We need n+1 rows ` `    ``# as the table is consturcted in bottom up ` `    ``# manner using the base case (n = 0) ` `    ``# Initialize all table values as 0 ` `    ``table ``=``[``0``] ``*` `(n ``+` `1``) ` ` `  `    ``# Base case (If given value is 0) ` `    ``# Only 1 way to get 0 (select no integer) ` `    ``table[``0``] ``=` `1` ` `  `    ``# Pick all integer one by one and update the ` `    ``# table[] values after the index greater ` `    ``# than or equal to n ` `    ``for` `i ``in` `range``(``1``, n ): ` ` `  `        ``for` `j ``in` `range``(i , n ``+` `1``): ` ` `  `            ``table[j] ``+``=`  `table[j ``-` `i]             ` ` `  `    ``return` `table[n] ` ` `  `# driver program ` `def` `main(): ` ` `  `    ``n ``=` `7` ` `  `    ``print` `CountWays(n) ` ` `  `if` `__name__ ``=``=` `'__main__'``: ` `    ``main() ` ` `  `#This code is contributed by Neelam Yadav `

## C#

 `// Program to find the number of ways, n can be ` `// written as sum of two or more positive integers. ` `using` `System; ` ` `  `class` `GFG { ` `     `  `    ``// Returns number of ways to write n as sum of ` `    ``// two or more positive integers ` `    ``static` `int` `countWays(``int` `n) ` `    ``{ ` `         `  `        ``// table[i] will be storing the number of ` `        ``// solutions for value i. We need n+1 rows ` `        ``// as the table is consturcted in bottom up ` `        ``// manner using the base case (n = 0) ` `        ``int` `[]table = ``new` `int``[n+1]; ` `      `  `        ``// Initialize all table values as 0 ` `        ``for``(``int` `i = 0; i < table.Length; i++) ` `            ``table[i] = 0; ` `      `  `        ``// Base case (If given value is 0) ` `        ``table = 1; ` `      `  `        ``// Pick all integer one by one and update the ` `        ``// table[] values after the index greater ` `        ``// than or equal to n ` `        ``for` `(``int` `i = 1; i < n; i++) ` `            ``for` `(``int` `j = i; j <= n; j++) ` `                ``table[j] += table[j-i]; ` `      `  `        ``return` `table[n]; ` `    ``} ` `     `  `    ``//driver code ` `    ``public` `static` `void` `Main() ` `    ``{ ` `        ``int` `n = 7; ` `         `  `        ``Console.Write(countWays(n)); ` `    ``} ` `} ` ` `  `// This code is contributed by Anant Agarwal. `

## PHP

 ` `

Output:

```14
```

Time complexity O(n2)

My Personal Notes arrow_drop_up

Improved By : chitranayal

Article Tags :
Practice Tags :

3

Please write to us at contribute@geeksforgeeks.org to report any issue with the above content.