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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.
Examples : 

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.

Algorithm: 

Step 1: Define a function named findCombinationsUtil which takes 4 parameters – arr[], index, num, and reducedNum.
Step 2: If the reducedNum is less than 0, return.
Step 3: If the reducedNum is equal to 0, print the array arr[] till index-1 and return.
Step 4: Find the previous element stored in arr[]. If index is 0, then prev = 1, else prev = arr[index – 1].
Step 5: Run a loop from prev to num.                                                                                                                                                                                
Step 6: For each value k in the loop:                                                                                                                                                                                                a. Set the next element of arr[] as k, i.e., arr[index] = k.                                                                                                                                                      b. Call the findCombinationsUtil function recursively with parameters arr[], index+1, num, and reducedNum-k.


Below is implementation of above idea :  

Java

// Java program to find out
// all combinations of positive
// numbers that add upto given
// number
import java.io.*;
 
class GFG
{
    /* arr - array to store the
    combination
    index - next location in array
    num - given number
    reducedNum - reduced number */
static 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++)
                System.out.print (arr[i] + " ");
            System.out.println();
        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() */
static void findCombinations(int n)
{
    // array to store the combinations
    // It can contain max n elements
    int arr[] = new int [n];
 
    // find all combinations
    findCombinationsUtil(arr, 0, n, n);
}
 
// Driver code
public static void main (String[] args)
{
    int n = 5;
    findCombinations(n);
}
}
 
// This code is contributed
// by akt_mit

                    

C++

// 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;
}

                    

Python3

# Python3 program to find out all
# combinations of positive
# numbers that add upto given number
 
# arr - array to store the combination
# index - next location in array
# num - given number
# reducedNum - reduced number
def findCombinationsUtil(arr, index, num,
                              reducedNum):
 
    # Base condition
    if (reducedNum < 0):
        return
 
    # If combination is
    # found, print it
    if (reducedNum == 0):
 
        for i in range(index):
            print(arr[i], end = " ")
        print("")
        return
 
    # Find the previous number stored in arr[].
    # It helps in maintaining increasing order
    prev = 1 if(index == 0) else arr[index - 1]
 
    # note loop starts from previous
    # number i.e. at array location
    # index - 1
    for k in range(prev, num + 1):
         
        # 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()
def findCombinations(n):
     
    # array to store the combinations
    # It can contain max n elements
    arr = [0] * n
 
    # find all combinations
    findCombinationsUtil(arr, 0, n, n)
 
# Driver code
n = 5;
findCombinations(n);
 
# This code is contributed by mits

                    

C#

// C# program to find out all
// combinations of positive numbers
// that add upto given number
using System;
 
class GFG
{
 
/* arr - array to store the
combination
index - next location in array
num - given number
reducedNum - reduced number */
static 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++)
            Console.Write (arr[i] + " ");
            Console.WriteLine();
        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() */
static void findCombinations(int n)
{
    // array to store the combinations
    // It can contain max n elements
    int []arr = new int [n];
 
    // find all combinations
    findCombinationsUtil(arr, 0, n, n);
}
 
// Driver code
static public void Main ()
{
    int n = 5;
    findCombinations(n);
}
}
 
// This code is contributed
// by akt_mit

                    

PHP

<?php
// PHP program to find out all
// combinations of positive
// numbers that add upto given number
 
/* arr - array to store the combination
    index - next location in array
    num - given number
    reducedNum - reduced number */
function findCombinationsUtil($arr, $index,
                              $num, $reducedNum)
{
    // Base condition
    if ($reducedNum < 0)
        return;
 
    // If combination is
    // found, print it
    if ($reducedNum == 0)
    {
        for ($i = 0; $i < $index; $i++)
            echo $arr[$i] , " ";
        echo "\n";
        return;
    }
 
    // Find the previous number
    // stored in arr[] It helps
    // in maintaining increasing order
    $prev = ($index == 0) ? 1 : $arr[$index - 1];
 
    // note loop starts from previous
    // number i.e. at array location
    // index - 1
    for ($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() */
function findCombinations($n)
{
    // array to store the combinations
    // It can contain max n elements
    $arr = array();
 
    //find all combinations
    findCombinationsUtil($arr, 0, $n, $n);
}
 
// Driver code
$n = 5;
findCombinations($n);
 
// This code is contributed by ajit
?>

                    

Javascript

<script>
 
// Javascript program to find out
// all combinations of positive
// numbers that add upto given
// number
 
    /* arr - array to store the
    combination
    index - next location in array
    num - given number
    reducedNum - reduced number */
function findCombinationsUtil(arr, index,
                                 num, reducedNum)
{
    // Base condition
    if (reducedNum < 0)
        return;
  
    // If combination is
    // found, print it
    if (reducedNum == 0)
    {
        for (let i = 0; i < index; i++)
               document.write (arr[i] + " ");
           document.write("<br/>");
        return;
    }
  
    // Find the previous number
    // stored in arr[]. It helps
    // in maintaining increasing
    // order
    let prev = (index == 0) ?
                          1 : arr[index - 1];
  
    // note loop starts from
    // previous number i.e. at
    // array location index - 1
    for (let 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() */
function findCombinations(n)
{
    // array to store the combinations
    // It can contain max n elements
    let arr = [];
  
    // find all combinations
    findCombinationsUtil(arr, 0, n, n);
}
 
// Driver Code
 
    let n = 5;
    findCombinations(n);      
                       
</script>

                    

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.

 

Time Complexity : O(2^n)
Auxiliary Space : O(n)



Last Updated : 10 Apr, 2023
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