Number of non-negative integral solutions of sum equation
Given a number n (number of variables) and val (sum of the variables), find out how many such nonnegative integral solutions are possible.
Examples :
Input : n = 5, val = 1 Output : 5 Explanation: x1 + x2 + x3 + x4 + x5 = 1 Number of possible solution are : (0 0 0 0 1), (0 0 0 1 0), (0 0 1 0 0), (0 1 0 0 0), (1 0 0 0 0) Total number of possible solutions are 5 Input : n = 5, val = 4 Output : 70 Explanation: x1 + x2 + x3 + x4 + x5 = 4 Number of possible solution are: (1 1 1 1 0), (1 0 1 1 1), (0 1 1 1 1), (2 1 0 0 1), (2 2 0 0 0)........ so on...... Total numbers of possible solutions are 70
Asked in: Microsoft Interview
1. Make a recursive function call to countSolutions(int n, int val)
2. Call this Solution function countSolutions(n-1, val-i) until n = 1 and val >=0 and then return 1.
Below is the implementation of above approach:
C++
// CPP program to find the numbers// of non negative integral solutions#include<iostream>using namespace std;// return number of non negative// integral solutionsint countSolutions(int n, int val){ // initialize total = 0 int total = 0; // Base Case if n = 1 and val >= 0 // then it should return 1 if (n == 1 && val >=0) return 1; // iterate the loop till equal the val for (int i = 0; i <= val; i++){ // total solution of equations // and again call the recursive // function Solutions(variable,value) total += countSolutions(n-1, val-i); } // return the total no possible solution return total;}// driver codeint main(){ int n = 5; int val = 20; cout<<countSolutions(n, val);}//This code is contributed by Smitha Dinesh Semwal |
Java
// Java program to find the numbers// of non negative integral solutionsclass GFG { // return number of non negative // integral solutions static int countSolutions(int n, int val) { // initialize total = 0 int total = 0; // Base Case if n = 1 and val >= 0 // then it should return 1 if (n == 1 && val >= 0) return 1; // iterate the loop till equal the val for (int i = 0; i <= val; i++) { // total solution of equations // and again call the recursive // function Solutions(variable, value) total += countSolutions(n - 1, val - i); } // return the total no possible solution return total; } // Driver code public static void main(String[] args) { int n = 5; int val = 20; System.out.print(countSolutions(n, val)); }}// This code is contributed by Anant Agarwal. |
Python3
# Python3 program to find the numbers# of non negative integral solutions# return number of non negative# integral solutionsdef countSolutions(n, val): # initialize total = 0 total = 0 # Base Case if n = 1 and val >= 0 # then it should return 1 if n == 1 and val >=0: return 1 # iterate the loop till equal the val for i in range(val+1): # total solution of equations # and again call the recursive # function Solutions(variable,value) total += countSolutions(n-1, val-i) # return the total no possible solution return total# driver coden = 5val = 20print(countSolutions(n, val)) |
C#
// C# program to find the numbers// of non negative integral solutionsusing System;class GFG { // return number of non negative // integral solutions static int countSolutions(int n, int val) { // initialize total = 0 int total = 0; // Base Case if n = 1 and val >= 0 // then it should return 1 if (n == 1 && val >= 0) return 1; // iterate the loop till equal the val for (int i = 0; i <= val; i++) { // total solution of equations // and again call the recursive // function Solutions(variable, value) total += countSolutions(n - 1, val - i); } // return the total no possible solution return total; } // Driver code public static void Main() { int n = 5; int val = 20; Console.WriteLine(countSolutions(n, val)); }}// This code is contributed by Anant vt_m. |
PHP
<?php// PHP program to find the numbers// of non negative integral solutions// return number of non negative// integral solutionsfunction countSolutions($n, $val){ // initialize total = 0 $total = 0; // Base Case if n = 1 and // val >= 0 then it should // return 1 if ($n == 1 && $val >=0) return 1; // iterate the loop // till equal the val for ($i = 0; $i <= $val; $i++) { // total solution of equations // and again call the recursive // function Solutions(variable,value) $total += countSolutions($n - 1, $val - $i); } // return the total // no possible solution return $total;}// Driver Code$n = 5;$val = 20;echo countSolutions($n, $val);// This code is contributed by nitin mittal.?> |
Javascript
<script>// JavaScript program to find the numbers// of non negative integral solutions // return number of non negative // integral solutions function countSolutions(n, val) { // initialize total = 0 let total = 0; // Base Case if n = 1 and val >= 0 // then it should return 1 if (n == 1 && val >= 0) return 1; // iterate the loop till equal the val for (let i = 0; i <= val; i++) { // total solution of equations // and again call the recursive // function Solutions(variable, value) total += countSolutions(n - 1, val - i); } // return the total no possible solution return total; }// Driver code let n = 5; let val = 20; document.write(countSolutions(n, val)); </script> |
Output :
10626
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