# 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
```

## 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 ` `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<

## 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

 `= 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 :

`10626`

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Improved By : nitin mittal, Akanksha_Rai