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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