# Find all combinations that add upto given number

Given a positive number, find out all combinations of positive numbers that adds upto that number. The program should print only combinations, not permutations. For example, for input 3, either 1, 2 or 2, 1 should be printed.

For example,

```Input: N = 3
Output:
1 1 1
1 2
3

Input: N = 5
Output:
1 1 1 1 1
1 1 1 2
1 1 3
1 2 2
1 4
2 3
5```

We strongly recommend you to minimize your browser and try this yourself first.

The idea is to use recursion. We use an array to store combinations and we recursively fill the array and recurse with reduced number. The invariant used in the solution is that each combination will always be stored in increasing order of elements involved. That way we can avoid printing permutations.

Below is C++ implementation of above idea :

`// C++ program to find out all combinations of`
`// positive numbers that add upto given number`
`#include <iostream>`
`using` `namespace` `std;`
`/*    arr - array to store the combination`
`    ``index - next location in array`
`    ``num - given number`
`    ``reducedNum - reduced number */`
`void` `findCombinationsUtil(``int` `arr[], ``int` `index,`
`                       ``int` `num, ``int` `reducedNum)`
`{`
`    ``// Base condition`
`    ``if` `(reducedNum < 0)`
`        ``return``;`
`    ``// If combination is found, print it`
`    ``if` `(reducedNum == 0)`
`    ``{`
`        ``for` `(``int` `i = 0; i < index; i++)`
`            ``cout << arr[i] << ``" "``;`
`        ``cout << endl;`
`        ``return``;`
`    ``}`
`    ``// Find the previous number stored in arr[]`
`    ``// It helps in maintaining increasing order`
`    ``int` `prev = (index == 0)? 1 : arr[index-1];`
`    ``// note loop starts from previous number`
`    ``// i.e. at array location index - 1`
`    ``for` `(``int` `k = prev; k <= num ; k++)`
`    ``{`
`        ``// next element of array is k`
`        ``arr[index] = k;`
`        ``// call recursively with reduced number`
`        ``findCombinationsUtil(arr, index + 1, num,`
`                                 ``reducedNum - k);`
`    ``}`
`}`
`/* Function to find out all combinations of`
`   ``positive numbers that add upto given number.`
`   ``It uses findCombinationsUtil() */`
`void` `findCombinations(``int` `n)`
`{`
`    ``// array to store the combinations`
`    ``// It can contain max n elements`
`    ``int` `arr[n];`
`    ``//find all combinations`
`    ``findCombinationsUtil(arr, 0, n, n);`
`}`
`// Driver code`
`int` `main()`
`{`
`    ``int` `n = 5;`
`    ``findCombinations(n);`
`    ``return` `0;`
`}`

Output:

```1 1 1 1 1
1 1 1 2
1 1 3
1 2 2
1 4
2 3
5
```

Exercise: Modify above solution to consider only distinct elements in a combination.

Disclaimer: This does not belong to TechCodeBit, its an article taken from the below
source and credits.