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Ways of dividing a group into two halves such that two elements are in different groups
  • Last Updated : 02 Feb, 2021

Given 2n girls and randomly divided into two subgroups each containing n girls. The task is to count the number of ways in which groups can be formed such that two beautiful girls are into different groups.
Example: 

Input:
Output:
Let group be r1, r2, b1, b2 where b1 and b2 are beautiful girls 
Groups are: ((r1, b1) (r2, b2)), ((r1, b2) (r2, b1)), ((r2, b2) (r1, b1)), ((r2, b1) (r1, b2))
Input:
Output: 40 

Approach: There are two ways in which the two beautiful girls lie in different groups and corresponding to each way the remaining (2n – 2) girls can be divided into two groups is {2n-2}_C_{n-1}
Hence total number of ways are 2 * {2n-2}_C_{n-1}
Implementation Code :  

C++




// CPP Program to count
// Number of ways in which two
// Beautiful girls are in different group
#include <bits/stdc++.h>
using namespace std;
 
// This function will
// return the factorial of a given number
int factorial(int n)
{
    int result = 1;
    for (int i = 1; i <= n; i++)
        result = result * i;
    return result;
}
 
// This function will calculate nCr of given
// n and r
int nCr(int n, int r)
{
    return factorial(n) / (factorial(r) * factorial(n - r));
}
 
// This function will
// Calculate number of ways
int calculate_result(int n)
{
    int result = 2 * nCr((n - 2), (n / 2 - 1));
    return result;
}
 
// Driver Code
int main(void)
{
    int a = 2, b = 4;
    cout << calculate_result(2 * a) << endl;
    cout << calculate_result(2 * b) << endl;
 
    return 0;
}

Java




   //Java Program to count
// Number of ways in which two
// Beautiful girls are in different group
 
import java.io.*;
 
class GFG {
 
// This function will
// return the factorial of a given number
static int factorial(int n)
{
    int result = 1;
    for (int i = 1; i <= n; i++)
        result = result * i;
    return result;
}
 
// This function will calculate nCr of given
// n and r
static int nCr(int n, int r)
{
    return factorial(n) / (factorial(r) * factorial(n - r));
}
 
// This function will
// Calculate number of ways
static int calculate_result(int n)
{
    int result = 2 * nCr((n - 2), (n / 2 - 1));
    return result;
}
 
// Driver Code
 
    public static void main (String[] args) {
        int a = 2, b = 4;
    System.out.println( calculate_result(2 * a));
    System.out.print(calculate_result(2 * b));
 
    }
}
 
// This code is contributed by inder_verma..

Python3




# Python3 Program to count
# Number of ways in which two
# Beautiful girls are in different group
 
# This function will
# return the factorial of a
# given number
def factorial(n) :
 
    result = 1
    for i in range(1, n + 1) :
        result *= i
         
    return result
 
# This function will calculate nCr of given
# n and r
def nCr(n, r) :
 
    return (factorial(n) // (factorial(r)
            * factorial(n - r)))
 
 
# This function will
# Calculate number of ways
def calculate_result(n) :
 
    result = 2 * nCr((n -2), (n // 2 - 1))
 
    return result
 
 
# Driver code
if __name__ == "__main__" :
 
    a, b = 2, 4
    print(calculate_result(2 * a))
    print(calculate_result(2 * b))
 
# This code is contributed by
# ANKITRAI1

C#




//C# Program to count
// Number of ways in which two
// Beautiful girls are in different groupusing System;
 
using System;
 
public class GFG {
  
// This function will
// return the factorial of a given number
static int factorial(int n)
{
    int result = 1;
    for (int i = 1; i <= n; i++)
        result = result * i;
    return result;
}
  
// This function will calculate nCr of given
// n and r
static int nCr(int n, int r)
{
    return factorial(n) / (factorial(r) * factorial(n - r));
}
  
// This function will
// Calculate number of ways
static int calculate_result(int n)
{
    int result = 2 * nCr((n - 2), (n / 2 - 1));
    return result;
}
  
// Driver Code
  
    public static void Main () {
        int a = 2, b = 4;
    Console.WriteLine( calculate_result(2 * a));
    Console.Write(calculate_result(2 * b));
  
    }
}
  
// This code is contributed by Subhadeep

PHP




<?php
// PHP Program to count Number
// of ways in which two Beautiful
// girls are in different group
 
// This function will return
// the factorial of a given number
function factorial($n)
{
    $result = 1;
    for ($i = 1; $i <= $n; $i++)
        $result = $result * $i;
    return $result;
}
 
// This function will calculate
// nCr of given n and r
function nCr($n, $r)
{
    return factorial($n) / (factorial($r) *
                            factorial($n - $r));
}
 
// This function will
// Calculate number of ways
function calculate_result($n)
{
    $result = 2 * nCr(($n - 2),
                      ($n / 2 - 1));
    return $result;
}
 
// Driver Code
$a = 2;
$b = 4;
echo calculate_result(2 * $a) . "\n";
echo calculate_result(2 * $b) . "\n";
 
// This Code is contributed by mits
?>
Output: 
4
40

 

Time Complexity: O(N )

Auxiliary Space: O(1)

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