# Number of ways to arrange 2*N persons on the two sides of a table with X and Y persons on opposite sides

Given three integers **N**, **X** and **Y**. The task is to find the number of ways to arrange** 2*N** persons along two sides of a table with **N** number of chairs on each side such that **X** persons are on one side and **Y** persons are on the opposite side.

**Note:** Both X and Y are less than or equals to N.

**Examples:**

Input :N = 5, X = 4, Y = 2

Output :57600

Explanation :

The total number of person 10. X men on one side and Y on other side, then 10 – 4 – 2 = 4 persons are left. We can choose 5 – 4 = 1 of them on one side in ways and the remaining persons will automatically sit on the other side. On each side arrangement is done in 5! ways. The number of ways is .5!5!

Input :N = 3, X = 1, Y = 2

Output :108

**Approach :**

The total number of person 2*N. Let call both the sides as A and B. X men on side A and Y on side B, then 2*N – X – Y persons are left. We can choose N-X of them for side A in ways and the remaining persons will automatically sit on the other side B. On each side arrangement is done in N! ways. The number of ways to arrange** 2*N** persons along two sides of a table is .N!N!

Below is the implementation of the above approach :

## C++

`#include <bits/stdc++.h> ` `using` `namespace` `std; ` ` ` `// Function to find factorial of a number ` `int` `factorial(` `int` `n) ` `{ ` ` ` `if` `(n <= 1) ` ` ` `return` `1; ` ` ` `return` `n * factorial(n - 1); ` `} ` ` ` `// Function to find nCr ` `int` `nCr(` `int` `n, ` `int` `r) ` `{ ` ` ` `return` `factorial(n) / (factorial(n - r) * factorial(r)); ` `} ` ` ` ` ` `// Function to find the number of ways to arrange 2*N persons ` `int` `NumberOfWays(` `int` `n, ` `int` `x, ` `int` `y) ` `{ ` ` ` `return` `nCr(2*n-x-y, n-x) * factorial(n) * factorial(n); ` `} ` ` ` ` ` `// Driver code ` `int` `main() ` `{ ` ` ` `int` `n = 5, x = 4, y = 2; ` ` ` ` ` `// Function call ` ` ` `cout << NumberOfWays(n, x, y); ` ` ` ` ` `return` `0; ` `} ` |

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## Java

`// Java implementation for the above approach ` `import` `java.util.*; ` `import` `java.lang.*; ` `import` `java.io.*; ` ` ` `class` `GFG ` `{ ` ` ` ` ` `// Function to returns factorial of n ` ` ` `static` `int` `factorial(` `int` `n) ` ` ` `{ ` ` ` `if` `(n <= ` `1` `) ` ` ` `return` `1` `; ` ` ` `return` `n * factorial(n - ` `1` `); ` ` ` `} ` ` ` ` ` `// Function to find nCr ` ` ` `static` `int` `nCr(` `int` `n, ` `int` `r) ` ` ` `{ ` ` ` `return` `factorial(n) / (factorial(n - r) * ` ` ` `factorial(r)); ` ` ` `} ` ` ` ` ` `// Function to find the number of ways ` ` ` `// to arrange 2*N persons ` ` ` `static` `int` `NumberOfWays(` `int` `n, ` `int` `x, ` `int` `y) ` ` ` `{ ` ` ` `return` `nCr(` `2` `* n - x - y, n - x) * ` ` ` `factorial(n) * factorial(n); ` ` ` `} ` ` ` ` ` `// Driver code ` ` ` `public` `static` `void` `main (String[] args) ` ` ` `throws` `java.lang.Exception ` ` ` `{ ` ` ` `int` `n = ` `5` `, x = ` `4` `, y = ` `2` `; ` ` ` ` ` `// Function call ` ` ` `System.out.println(NumberOfWays(n, x, y)); ` ` ` `} ` `} ` ` ` `// This code is contributed by Nidhiva ` |

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## Python3

`# Python3 implementation for the above approach ` ` ` `# Function to find factorial of a number ` `def` `factorial(n): ` ` ` ` ` `if` `(n <` `=` `1` `): ` ` ` `return` `1` `; ` ` ` `return` `n ` `*` `factorial(n ` `-` `1` `); ` ` ` `# Function to find nCr ` `def` `nCr(n, r): ` ` ` ` ` `return` `(factorial(n) ` `/` ` ` `(factorial(n ` `-` `r) ` `*` `factorial(r))); ` ` ` `# Function to find the number of ways ` `# to arrange 2*N persons ` `def` `NumberOfWays(n, x, y): ` ` ` ` ` `return` `(nCr(` `2` `*` `n ` `-` `x ` `-` `y, n ` `-` `x) ` `*` ` ` `factorial(n) ` `*` `factorial(n)); ` ` ` `# Driver code ` `n, x, y ` `=` `5` `, ` `4` `, ` `2` `; ` ` ` `# Function call ` `print` `(` `int` `(NumberOfWays(n, x, y))); ` ` ` `# This code is contributed by PrinciRaj1992 ` |

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## C#

`// C# implementation for the above approach ` `using` `System; ` ` ` `class` `GFG ` `{ ` ` ` ` ` `// Function to returns factorial of n ` ` ` `static` `int` `factorial(` `int` `n) ` ` ` `{ ` ` ` `if` `(n <= 1) ` ` ` `return` `1; ` ` ` `return` `n * factorial(n - 1); ` ` ` `} ` ` ` ` ` `// Function to find nCr ` ` ` `static` `int` `nCr(` `int` `n, ` `int` `r) ` ` ` `{ ` ` ` `return` `factorial(n) / (factorial(n - r) * ` ` ` `factorial(r)); ` ` ` `} ` ` ` ` ` `// Function to find the number of ways ` ` ` `// to arrange 2*N persons ` ` ` `static` `int` `NumberOfWays(` `int` `n, ` `int` `x, ` `int` `y) ` ` ` `{ ` ` ` `return` `nCr(2 * n - x - y, n - x) * ` ` ` `factorial(n) * factorial(n); ` ` ` `} ` ` ` ` ` `// Driver code ` ` ` `public` `static` `void` `Main(String[] args) ` ` ` `{ ` ` ` `int` `n = 5, x = 4, y = 2; ` ` ` ` ` `// Function call ` ` ` `Console.WriteLine(NumberOfWays(n, x, y)); ` ` ` `} ` `} ` ` ` `// This code is contributed by Princi Singh ` |

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## PHP

`<?php ` `// PHP implementation for the above approach ` ` ` `// Function to find factorial of a number ` `function` `factorial(` `$n` `) ` `{ ` ` ` `if` `(` `$n` `<= 1) ` ` ` `return` `1; ` ` ` `return` `$n` `* factorial(` `$n` `- 1); ` `} ` ` ` `// Function to find nCr ` `function` `nCr(` `$n` `, ` `$r` `) ` `{ ` ` ` `return` `factorial(` `$n` `) / (factorial(` `$n` `- ` `$r` `) * ` ` ` `factorial(` `$r` `)); ` `} ` ` ` `// Function to find the number of ways ` `// to arrange 2*N persons ` `function` `NumberOfWays(` `$n` `, ` `$x` `, ` `$y` `) ` `{ ` ` ` `return` `nCr(2 * ` `$n` `- ` `$x` `- ` `$y` `, ` `$n` `- ` `$x` `) * ` ` ` `factorial(` `$n` `) * factorial(` `$n` `); ` `} ` ` ` `// Driver code ` `$n` `= 5; ` `$x` `= 4; ` `$y` `= 2; ` ` ` `// Function call ` `echo` `(NumberOfWays(` `$n` `, ` `$x` `, ` `$y` `)); ` ` ` `// This code is contributed by Naman_garg. ` `?> ` |

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**Output:**

57600

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