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Friends Pairing Problem

Given n friends, each one can remain single or can be paired up with some other friend. Each friend can be paired only once. Find out the total number of ways in which friends can remain single or can be paired up.

Examples :

Input  : n = 3
Output : 4

Explanation
{1}, {2}, {3} : all single
{1}, {2, 3} : 2 and 3 paired but 1 is single.
{1, 2}, {3} : 1 and 2 are paired but 3 is single.
{1, 3}, {2} : 1 and 3 are paired but 2 is single.
Note that {1, 2} and {2, 1} are considered same.

f(n) = ways n people can remain single 
       or pair up.

For n-th person there are two choices:
1) n-th person remains single, we recur 
   for f(n - 1)
2) n-th person pairs up with any of the 
   remaining n - 1 persons. We get (n - 1) * f(n - 2)

Therefore we can recursively write f(n) as:
f(n) = f(n - 1) + (n - 1) * f(n - 2)

Since above recursive formula has overlapping subproblems, we can solve it using Dynamic Programming.

C++



// C++ program for solution of
// friends pairing problem
#include <bits/stdc++.h>
using namespace std;
  
// Returns count of ways n people
// can remain single or paired up.
int countFriendsPairings(int n)
{
    int dp[n + 1];
  
    // Filling dp[] in bottom-up manner using
    // recursive formula explained above.
    for (int i = 0; i <= n; i++) {
        if (i <= 2)
            dp[i] = i;
        else
            dp[i] = dp[i - 1] + (i - 1) * dp[i - 2];
    }
  
    return dp[n];
}
  
// Driver code
int main()
{
    int n = 4;
    cout << countFriendsPairings(n) << endl;
    return 0;
}

Java

// Java program for solution of
// friends pairing problem
import java.io.*;
  
class GFG {
  
    // Returns count of ways n people
    // can remain single or paired up.
    static int countFriendsPairings(int n)
    {
        int dp[] = new int[n + 1];
  
        // Filling dp[] in bottom-up manner using
        // recursive formula explained above.
        for (int i = 0; i <= n; i++) {
            if (i <= 2)
                dp[i] = i;
            else
                dp[i] = dp[i - 1] + (i - 1) * dp[i - 2];
        }
  
        return dp[n];
    }
  
    // Driver code
    public static void main(String[] args)
    {
        int n = 4;
        System.out.println(countFriendsPairings(n));
    }
}
  
// This code is contributed by vt_m

Python3

# Python program solution of
# friends pairing problem
  
# Returns count of ways
# n people can remain
# single or paired up.
def countFriendsPairings(n):
  
    dp = [0 for i in range(n + 1)]
  
    # Filling dp[] in bottom-up manner using
    # recursive formula explained above.
    for i in range(n + 1):
  
        if(i <= 2):
            dp[i] = i
        else:
            dp[i] = dp[i - 1] + (i - 1) * dp[i - 2]
  
    return dp[n]
  
# Driver code
n = 4
print(countFriendsPairings(n))
  
# This code is contributed
# by Soumen Ghosh.

C#

// C# program solution for
// friends pairing problem
using System;
  
class GFG {
  
    // Returns count of ways n people
    // can remain single or paired up.
    static int countFriendsPairings(int n)
    {
        int[] dp = new int[n + 1];
  
        // Filling dp[] in bottom-up manner using
        // recursive formula explained above.
        for (int i = 0; i <= n; i++) {
            if (i <= 2)
                dp[i] = i;
            else
                dp[i] = dp[i - 1] + (i - 1) * dp[i - 2];
        }
  
        return dp[n];
    }
  
    // Driver code
    public static void Main()
    {
        int n = 4;
        Console.Write(countFriendsPairings(n));
    }
}
  
// This code is contributed by nitin mittal.

PHP

<?php
// PHP program solution for 
// friends pairing problem
  
// Returns count of ways n people 
// can remain single or paired up.
function countFriendsPairings($n)
{
    $dp[$n + 1] = 0;
  
    // Filling dp[] in bottom-up 
    // manner using recursive formula 
    // explained above.
    for ($i = 0; $i <= $n; $i++)
    {
        if ($i <= 2)
        $dp[$i] = $i;
        else
        $dp[$i] = $dp[$i - 1] + 
                     ($i - 1) * 
                   $dp[$i - 2];
    }
  
    return $dp[$n];
}
  
// Driver code
$n = 4;
echo countFriendsPairings($n) ;
      
// This code is contributed 
// by nitin mittal.
?>


Output :

10

Time Complexity : O(n)
Auxiliary Space : O(n)

Another approach: (Using recursion)

C++

// C++ program for solution of friends
// pairing problem Using Recursion
#include <bits/stdc++.h>
using namespace std;
  
int dp[1000];
  
// Returns count of ways n people
// can remain single or paired up.
int countFriendsPairings(int n)
{
    if (dp[n] != -1)
        return dp[n];
  
    if (n > 2)
        return dp[n] = countFriendsPairings(n - 1) + 
                       (n - 1) * countFriendsPairings(n - 2);
    else
        return dp[n] = n;
}
  
// Driver code
int main()
{
    memset(dp, -1, sizeof(dp));
    int n = 4;
    cout << countFriendsPairings(n) << endl;
    // this code is contributed by Kushdeep Mittal
}

Java

// Java program for solution of friends
// pairing problem Using Recursion
  
class GFG {
    static int[] dp = new int[1000];
  
    // Returns count of ways n people
    // can remain single or paired up.
    static int countFriendsPairings(int n)
    {
        if (dp[n] != -1)
            return dp[n];
  
        if (n > 2)
            return dp[n] = countFriendsPairings(n - 1) + 
                           (n - 1) * countFriendsPairings(n - 2);
        else
            return dp[n] = n;
    }
  
    // Driver code
    public static void main(String[] args)
    {
        for (int i = 0; i < 1000; i++)
            dp[i] = -1;
        int n = 4;
        System.out.println(countFriendsPairings(n));
    }
}
  
// This code is contributed by Ita_c.

Python3

# Python3 program for solution of friends 
# pairing problem Using Recursion 
  
# Returns count of ways n people 
# can remain single or paired up. 
def countFriendsPairings(n): 
  
    dp = [-1] * 100
      
    if(dp[n] != -1): 
        return dp[n] 
  
    if(n > 2): 
  
        dp[n] = (countFriendsPairings(n - 1) + 
                (n - 1) * countFriendsPairings(n - 2)) 
        return dp[n]
  
    else:
        dp[n] =
        return dp[n]
      
# Driver Code
n = 4
print(countFriendsPairings(n))
  
# This code contributed by PrinciRaj1992

C#

// C# program for solution of friends
// pairing problem Using Recursion
using System;
  
class GFG {
    static int[] dp = new int[1000];
  
    // Returns count of ways n people
    // can remain single or paired up.
    static int countFriendsPairings(int n)
    {
        if (dp[n] != -1)
            return dp[n];
  
        if (n > 2)
            return dp[n] = countFriendsPairings(n - 1) + 
                           (n - 1) * countFriendsPairings(n - 2);
        else
            return dp[n] = n;
    }
  
    // Driver code
    static void Main()
    {
        for (int i = 0; i < 1000; i++)
            dp[i] = -1;
        int n = 4;
        Console.Write(countFriendsPairings(n));
    }
}
  
// This code is contributed by DrRoot_

PHP

<?php
// PHP program for solution of friends 
// pairing problem Using Recursion 
  
// Returns count of ways n people 
// can remain single or paired up. 
function countFriendsPairings($n
    $dp = array_fill(0, 1000, -1);
      
    if($dp[$n] != -1) 
    return $dp[$n]; 
  
    if($n > 2) 
    {
        $dp[$n] = countFriendsPairings($n - 1) + ($n - 1) * 
                  countFriendsPairings($n - 2); 
        return $dp[$n];
    }
    else
    {
        $dp[$n] = $n
        return $dp[$n];
    }
      
// Driver Code
$n = 4; 
echo countFriendsPairings($n
  
// This code is contributed by Ryuga
?>

Output :



10

Time Complexity : O(n)
Auxiliary Space : O(n)

Since above formula is similar to fibonacci number, we can optimize the space with an iterative solution.

Java

class GFG {
    // Returns count of ways n people
    // can remain single or paired up.
    static int countFriendsPairings(int n)
    {
        int a = 1, b = 2, c = 0;
        if (n <= 2) {
            return n;
        }
        for (int i = 3; i <= n; i++) {
            c = b + (i - 1) * a;
            a = b;
            b = c;
        }
        return c;
    }
  
    // Driver code
    public static void main(String[] args)
    {
        int n = 4;
        System.out.println(countFriendsPairings(n));
    }
}
  
// This code is contributed by Ravi Kasha.

Python3

# Returns count of ways n people
# can remain single or paired up.
def countFriendsPairings(n):
    a, b, c =1, 2, 0;
    if (n <= 2):
        return n;
    for i in range(3,n+1):
        c = b + (i - 1) * a;
        a = b;
        b = c;
    return c;
  
# Driver code
n = 4;
print(countFriendsPairings(n));
  
#This code contributed by Rajput-Ji

C#

using System;
  
class GFG 
    // Returns count of ways n people 
    // can remain single or paired up. 
    static int countFriendsPairings(int n) 
    
        int a = 1, b = 2, c = 0; 
        if (n <= 2) 
        
            return n; 
        
        for (int i = 3; i <= n; i++) 
        
            c = b + (i - 1) * a; 
            a = b; 
            b = c; 
        
        return c; 
    
  
    // Driver code 
    public static void Main(String[] args) 
    
        int n = 4; 
        Console.WriteLine(countFriendsPairings(n)); 
    
  
// This code has been contributed by 29AjayKumar

PHP

<?php
    // Returns count of ways n people
    // can remain single or paired up.
    function countFriendsPairings($n)
    {
        $a = 1;
        $b = 2;
        $c = 0;
        if ($n <= 2)
        {
            return $n;
        }
        for ($i = 3; $i <= $n; $i++)
        {
            $c = $b + ($i - 1) * $a;
            $a = $b;
            $b = $c;
        }
        return $c;
    }
  
    // Driver code
        $n = 4;
        print(countFriendsPairings($n));
  
// This code is contributed by mits
?>


Output :

10

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

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.



This article is attributed to GeeksforGeeks.org

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