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

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

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 solutions 
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 
int 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 solutions 
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(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 solutions 
def 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 code 
n = 5
val = 20
print(countSolutions(n, val)) 

C#

// C# program to find the numbers 
// of non negative integral solutions 
using 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 solutions 
function 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.  
?> 

Output :

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My Personal Notes

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

C++

Java

Python3

C#

PHP

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