# 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

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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++

`// Program to find the number of ways, n can be` `// written as sum of two or more positive integers.` `#include <bits/stdc++.h>` `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 constructed 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[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 program` `int` `main()` `{` ` ` `int` `n = 7;` ` ` `cout << countWays(n);` ` ` `return` `0;` `}` |

## 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 constructed 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 constructed 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 constructed 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[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()` ` ` `{` ` ` `int` `n = 7;` ` ` ` ` `Console.Write(countWays(n));` ` ` `}` `}` `// This code is contributed by Anant Agarwal.` |

## PHP

`<?php` `// 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` `function` `countWays(` `$n` `)` `{` ` ` `// table[i] will be storing the number of` ` ` `// solutions for value i. We need n+1 rows` ` ` `// as the table is constructed in bottom up` ` ` `// manner using the base case (n = 0)` ` ` `$table` `= ` `array_fill` `(0, ` `$n` `+ 1, NULL);` ` ` `// Base case (If given value is 0)` ` ` `$table` `[0] = 1;` ` ` `// Pick all integer one by one and update` ` ` `// the table[] values after the index` ` ` `// greater than or equal to n` ` ` `for` `(` `$i` `= 1; ` `$i` `< ` `$n` `; ` `$i` `++)` ` ` `for` `(` `$j` `= ` `$i` `; ` `$j` `<= ` `$n` `; ` `$j` `++)` ` ` `$table` `[` `$j` `] += ` `$table` `[` `$j` `- ` `$i` `];` ` ` `return` `$table` `[` `$n` `];` `}` `// Driver Code` `$n` `= 7;` `echo` `countWays(` `$n` `);` `// This code is contributed by ita_c` `?>` |

## Javascript

`<script>` ` ` ` ` `function` `countWays(n)` ` ` `{` ` ` `// table[i] will be storing the` ` ` `// number of solutions for value` ` ` `// i. We need n+1 rows as the` ` ` `// table is constructed in bottom` ` ` `// up manner using the base case` ` ` `// (n = 0)` ` ` `let table = ` `new` `Array(n + 1);` ` ` ` ` `// Initialize all table values as 0` ` ` `for` `(let i = 0; i < n + 1; i++)` ` ` `{` ` ` `table[i]=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` `(let i = 1; i < n; i++)` ` ` `for` `(let j = i; j <= n; j++)` ` ` `table[j] += table[j - i];` ` ` ` ` `return` `table[n];` ` ` `}` ` ` ` ` `let n = 7;` ` ` `document.write(countWays(n));` ` ` ` ` `// This code is contributed by avanitrachhadiya2155` `</script>` |

Output:

14

Time complexity O(n^{2})

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