# Permutations of n things taken r at a time with k things together

Given n, r and K. The task is to find the number of permutations of different things taken at a time such that specific things always occur together.

Examples:

Input : n = 8, r = 5, k = 2
Output : 960

Input : n = 6, r = 2, k = 2
Output : 2


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

Approach:

1. A bundle of specific things can be put in r places in (r – k + 1) ways .
2. k specific things in the bundle can be arranged themselves into k! ways.
3. Now (n – k) things will be arranged in (r – k) places in ways.

Thus, using the fundamental principle of counting, the required number of permutations will be:

Below is the implementation of the above approach:

## C++

 // CPP program to find the number of permutations of  // n different things taken r at a time  // with k things grouped together     #include  using namespace std;     // Function to find factorial  // of a number  int factorial(int n)  {      int fact = 1;         for (int i = 2; i <= n; i++)          fact = fact * i;         return fact;  }     // Function to calculate p(n, r)  int npr(int n, int r)  {      int pnr = factorial(n) / factorial(n - r);         return pnr;  }     // Function to find the number of permutations of  // n different things taken r at a time  // with k things grouped together  int countPermutations(int n, int r, int k)  {      return factorial(k) * (r - k + 1) * npr(n - k, r - k);  }     // Driver code  int main()  {      int n = 8;      int r = 5;      int k = 2;         cout << countPermutations(n, r, k);         return 0;  }

## Java

 // Java program to find the number of permutations of  // n different things taken r at a time  // with k things grouped together     class GFG{  // Function to find factorial  // of a number  static int factorial(int n)  {      int fact = 1;         for (int i = 2; i <= n; i++)          fact = fact * i;         return fact;  }     // Function to calculate p(n, r)  static int npr(int n, int r)  {      int pnr = factorial(n) / factorial(n - r);         return pnr;  }     // Function to find the number of permutations of  // n different things taken r at a time  // with k things grouped together  static int countPermutations(int n, int r, int k)  {      return factorial(k) * (r - k + 1) * npr(n - k, r - k);  }     // Driver code  public static void main(String[] args)  {      int n = 8;      int r = 5;      int k = 2;         System.out.println(countPermutations(n, r, k));  }  }  // this code is contributed by mits

## Python3

 # Python3 program to find the number of permutations of   # n different things taken r at a time   # with k things grouped together      # def to find factorial   # of a number   def factorial(n):           fact = 1;          for i in range(2,n+1):           fact = fact * i;          return fact;          # def to calculate p(n, r)   def npr(n, r):           pnr = factorial(n) / factorial(n - r);          return pnr;          # def to find the number of permutations of   # n different things taken r at a time   # with k things grouped together   def countPermutations(n, r, k):           return int(factorial(k) * (r - k + 1) * npr(n - k, r - k));          # Driver code   n = 8;   r = 5;   k = 2;      print(countPermutations(n, r, k));         # this code is contributed by mits

## C#

 // C# program to find the number of  // permutations of n different things  // taken r at a time with k things   // grouped together   using System;     class GFG  {         // Function to find factorial   // of a number   static int factorial(int n)   {       int fact = 1;          for (int i = 2; i <= n; i++)           fact = fact * i;          return fact;   }      // Function to calculate p(n, r)   static int npr(int n, int r)   {       int pnr = factorial(n) /                factorial(n - r);          return pnr;   }      // Function to find the number of   // permutations of n different   // things taken r at a time with   // k things grouped together   static int countPermutations(int n,                                int r, int k)   {       return factorial(k) * (r - k + 1) *                       npr(n - k, r - k);   }      // Driver code   static void Main()   {       int n = 8;       int r = 5;       int k = 2;          Console.WriteLine(countPermutations(n, r, k));   }   }      // This code is contributed by mits

## PHP

 

Output:

960


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