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# Count arrangements of N people around circular table such that K people always sit together

Given integers N and K, the task is to find the number of possible arrangements of N people around a circular table such that K people always sit together.

Note: As the answer can be very large return it modulo 109 + 7

Examples:

Input: N = 4, K = 3
Output: 6
Explanation: If 3 people always sit together (say 1, 2 and 3) then the possible arrangements can be
{1, 2, 3, 4}, {1, 3, 2, 4}, {2, 1, 3, 4}, {2, 3, 1, 4}, {3, 2, 1, 4}, {3, 1, 2, 4}.
As there is no start or end in a circular arrangement and
the arrangements are based on relative positions, so we can consider
4 is always fixed in the last position.

Input: N = 8, K = 3
Output: 720

Approach: This problem can be solved based on the following mathematical idea:

In a circular arrangement there is no starting or ending and the arrangements are based on relative positions.
So it can be considered that any one of the person is always sitting in a fixed position and all other positions are relative to that position.
So total possible arrangements of N people around a circular table is (N-1)!

As K people will always sit together, they can be considered a group or as a single unit.
So total unit now X = (N – K + 1). Total possible arrangements of these many people are:
(X – 1)! = (N – K)!

The people of this single group can be arranged in K! ways for each of the possible arrangements.
therefore total possible arrangements = K! * (N – K)!

Follow the below steps to implement the above approach.

Below is the implementation of the above approach.

## C++

 `// C++ code to implement the approach``#include ``using` `namespace` `std;` `const` `int` `mod = 1000000007;` `// Function to compute factorial of a number``int` `fac(``int` `f) {``    ``int` `fac = 1;``    ``for` `(``int` `i = 1; i <= f; i++)``        ``fac = (fac * i) % mod;``    ``return` `fac;``}` `// Function to find number of ways``int` `Ways(``int` `n, ``int` `k)``{``  ` `    ``// Find (n-k) factorial and k factorial``    ``// Return product of these two``    ``return` `(fac(n - k) * fac(k)) % mod;``}``  ` `// Driver code ``int` `main() {``    ``int` `N = 8;``    ``int` `K = 3;`  `    ``cout <<    Ways(N, K);``    ``return` `0;``}` `// This code is contributed by ANKITKUMAR34.`

## Java

 `// Java code for the above approach` `import` `java.io.*;` `class` `GFG {` `    ``static` `int` `mod = ``1000000007``;` `    ``// Function to compute factorial of a number``    ``static` `int` `fac(``int` `f)``    ``{``        ``int` `fac = ``1``;``        ``for` `(``int` `i = ``1``; i <= f; i++)``            ``fac = (fac * i) % mod;``        ``return` `fac;``    ``}` `    ``// Function to find number of ways``    ``static` `int` `Ways(``int` `n, ``int` `k)``    ``{` `        ``// Find (n-k) factorial and k factorial``        ``// Return product of these two``        ``return` `(fac(n - k) * fac(k)) % mod;``    ``}` `    ``public` `static` `void` `main(String[] args)``    ``{``        ``int` `N = ``8``;``        ``int` `K = ``3``;` `        ``System.out.println(Ways(N, K));``    ``}``}` `// This code is contributed by Potta Lokesh`

## Python3

 `# Python code to implement the approach` `mod ``=` `1000000007` `# Function to compute factorial of a number``def` `fac(f):``    ``fac ``=` `1``    ``for` `i ``in` `range``(``2``, f ``+` `1``):``        ``fac ``=` `(fac ``*` `i) ``%` `mod``    ``return` `fac   ` `# Function to find number of ways``def` `Ways(n, k):``    ``# Find (n-k) factorial and k factorial``    ``# Return product of these two``    ``return` `(fac(n ``-` `k) ``*` `fac(k)) ``%` `mod` `  ` `# Driver code``if` `__name__ ``=``=` `'__main__'``:``    ``N ``=` `8``    ``K ``=` `3``    ``print``(Ways(N, K));`

## C#

 `// C# code to implement the above approach``using` `System;` `public` `class` `GFG``{` `  ``static` `int` `mod = 1000000007;` `  ``// Function to compute factorial of a number``  ``static` `int` `fac(``int` `f) {``    ``int` `fac = 1;``    ``for` `(``int` `i = 1; i <= f; i++)``      ``fac = (fac * i) % mod;``    ``return` `fac;``  ``}` `  ``// Function to find number of ways``  ``static` `int` `Ways(``int` `n, ``int` `k)``  ``{` `    ``// Find (n-k) factorial and k factorial``    ``// Return product of these two``    ``return` `(fac(n - k) * fac(k)) % mod;``  ``}` `  ``// Driver code ``  ``static` `public` `void` `Main (){``    ``int` `N = 8;``    ``int` `K = 3;` `    ``Console.WriteLine(Ways(N, K));``  ``}``}` `// This code is contributed by AnkThon`

## Javascript

 ``

Output

`720`

Time Complexity: O(N)
Auxiliary Space: O(1)