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
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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// numberimport 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 givennumber. 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 codepublic 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 codeint 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 coden = 5;findCombinations(n);# This code is contributed by mits |
C#
// C# program to find out all// combinations of positive numbers// that add upto given numberusing 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 allcombinations of positivenumbers that add upto givennumber. 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> |
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.
This article is contributed by Aditya Goel. If you like neveropen and would like to contribute, you can also write an article and mail your article to review-team@geeksforgeeks.org. See your article appearing on the neveropen main page and help other Geeks.
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Time Complexity : O(2^n)
Auxiliary Space : O(n)
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