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

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

Examples:

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

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

Approach:

A bundle of $k$ specific things can be put in r places in (r – k + 1) ways .
k specific things in the bundle can be arranged themselves into k! ways.
Now (n – k) things will be arranged in (r – k) places in $n-k_{P_r-k}$ ways.

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

$Permutations = k! \times (r-k+1) \times $n-k_{P_r-k}$$

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 <bits/stdc++.h>
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

<?php
// PHP program to find the number 
// of permutations of n different 
// things taken r at a time
// with k things grouped together
 
// Function to find factorial
// of a number
function factorial($n)
{
    $fact = 1;
 
    for ($i = 2; $i <= $n; $i++)
        $fact = $fact * $i;
 
    return $fact;
}
 
// Function to calculate p(n, r)
function npr($n, $r)
{
    $pnr = factorial($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
function countPermutations($n, $r, $k)
{
    return factorial($k) * ($r - $k + 1) * 
                   npr($n - $k, $r - $k);
}
 
// Driver code
$n = 8;
$r = 5;
$k = 2;
 
echo countPermutations($n, $r, $k);
 
// This code is contributed by mits
?>

Javascript

<script>
 
// JavaScript program to find the number of permutations of 
// n different things taken r at a time 
// with k things grouped together 
 
// Function to find factorial 
// of a number 
function factorial(n) 
{ 
    let fact = 1; 
 
    for (let i = 2; i <= n; i++) 
        fact = fact * i; 
 
    return fact; 
} 
 
// Function to calculate p(n, r) 
function npr(n, r) 
{ 
    let pnr = Math.floor(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 
 
function countPermutations(n, r, k) 
{ 
    return factorial(k) * (r - k + 1) * npr(n - k, r - k); 
} 
 
// Driver code 
    let n = 8; 
    let r = 5; 
    let k = 2; 
 
    document.write(countPermutations(n, r, k)); 
 
// This code is contributed by Surbhi Tyagi.
 
</script>

Output:

Time Complexity: O(n)

Auxiliary Space: O(1)

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