# Number of integral solutions of the equation x1 + x2 +…. + xN = k

Given N and K. The task is to count the number of the integral solutions of a linear equation having N variable as given below:

x1 + x2+ x3…+ xN-1+…+xN = K

Examples:

```Input: N = 3, K = 3
Output: 10

Input: N = 2, K = 2
Output: 3
```

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

Approach: This problem can be solved using the concept of Permutation and Combination. Below are the direct formulas for finding non-negative and positive integral solutions respectively.

Number of non-negative integral solutions of equation x1 + x2 + …… + xn = k is given by (n+k-1)! / (n-1)!*k!.
Number of positive integral solutions of equation x1 + x2 + ….. + xn = k is given by (k-1)! / (n-1)! * (k-n)!.

Below is the implementation of above approach:

## c++

 `// C++ program for above implementation  ` `#include ` ` `  `using` `namespace` `std ; ` ` `  `int` `nCr(``int` `n, ``int` `r) ` `{ ` `    ``int` `fac = {1} ; ` `     `  `    ``for` `(``int` `i = 1 ; i < n + 1 ; i++) ` `    ``{ ` `        ``fac[i] = fac[i - 1] * i ; ` `    ``} ` `     `  `    ``int` `ans = fac[n] / (fac[n - r] *  ` `                        ``fac[r]) ; ` `    ``return` `ans ; ` `} ` ` `  `// Driver Code ` `int` `main() ` `{ ` `    ``int` `n = 3 ; ` `    ``int` `k = 3 ; ` `     `  `    ``int` `ans = nCr(n + k - 1 , k) +  ` `              ``nCr(k - 1, n - 1); ` `     `  `    ``cout << ans ; ` `     `  `    ``return` `0 ;  ` `} ` ` `  `// This code is contributed ` `// by ANKITRAI1 `

## Java

 `// Java program for above implementation ` `import` `java.io.*; ` ` `  `class` `GFG  ` `{ ` `static` `int` `nCr(``int` `n, ``int` `r) ` `{ ` `    ``int` `fac[] = ``new` `int``[``100``] ; ` `    ``for``(``int` `i = ``0``; i < n; i++) ` `    ``fac[i] = ``1``; ` `     `  `    ``for` `(``int` `i = ``1` `; i < n + ``1` `; i++) ` `    ``{ ` `        ``fac[i] = fac[i - ``1``] * i ; ` `    ``} ` `     `  `    ``int` `ans = fac[n] / (fac[n - r] *  ` `                        ``fac[r]); ` `    ``return` `ans ; ` `} ` ` `  `// Driver Code ` `public` `static` `void` `main (String[] args)  ` `{ ` `    ``int` `n = ``3` `; ` `    ``int` `k = ``3` `; ` `     `  `    ``int` `ans = nCr(n + k - ``1` `, k) +  ` `              ``nCr(k - ``1``, n - ``1``); ` `     `  `    ``System.out.println(ans) ; ` `} ` `} ` ` `  `// This code is contributed ` `// by anuj_67 `

## Python3

 `# Python implementation of  ` `# above approach ` ` `  `# Calculate nCr i.e binomial  ` `# cofficent nCr = n !/(r !*(n-r)!) ` `def` `nCr(n, r): ` ` `  `    ``# first find factorial  ` `    ``# upto n ` `    ``fac ``=` `list``() ` `    ``fac.append(``1``) ` `    ``for` `i ``in` `range``(``1``, n ``+` `1``): ` `        ``fac.append(fac[i ``-` `1``] ``*` `i) ` ` `  `    ``# use nCr formula ` `    ``ans ``=` `fac[n] ``/` `(fac[n ``-` `r] ``*` `fac[r]) ` `    ``return` `ans ` ` `  `# n = number of variables ` `n ``=` `3` ` `  `# sum of n variables = k ` `k ``=` `3` ` `  `# find number of solutions ` `ans ``=` `nCr(n ``+` `k ``-` `1``, k) ``+` `nCr(k ``-` `1``, n ``-` `1``) ` ` `  `print``(ans) ` ` `  `# This code is contributed ` `# by ChitraNayal `

## C#

 `// C# program for above implementation ` `using` `System;  ` ` `  `class` `GFG  ` `{ ` `static` `int` `nCr(``int` `n, ``int` `r) ` `{ ` `    ``int``[] fac = ``new` `int`` ; ` `    ``for``(``int` `i = 0; i < n; i++) ` `    ``fac[i] = 1; ` `     `  `    ``for` `(``int` `i = 1 ; i < n + 1 ; i++) ` `    ``{ ` `        ``fac[i] = fac[i - 1] * i ; ` `    ``} ` `     `  `    ``int` `ans = fac[n] / (fac[n - r] *  ` `                        ``fac[r]); ` `    ``return` `ans ; ` `} ` ` `  `// Driver Code ` `public` `static` `void` `Main ()  ` `{ ` `    ``int` `n = 3 ; ` `    ``int` `k = 3 ; ` `     `  `    ``int` `ans = nCr(n + k - 1 , k) +  ` `              ``nCr(k - 1, n - 1); ` `     `  `    ``Console.Write(ans) ; ` `} ` `} ` ` `  `// This code is contributed  ` `// by ChitraNayal `

## PHP

 ` `

Output:

```11.0
```

Applications of the above concepts:

1. Number of non-negative integral solutions of equation x1 + x2 +…+ xn = k is equal to the number of ways in which k identical balls can be distributed into N unique boxes.
2. Number of positive integral solutions of equation x1 + x2 + … + xn = k is equal to the number of ways in which k identical balls can be distributed into N unique boxes such that each box must contain at-least 1 ball.

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