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++

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// 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

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Java

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// 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.

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Python3

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# 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))

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C#

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// 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.

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PHP

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<?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. 
?>

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Output :

10626


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Improved By : nitin mittal, Akanksha_Rai