Ways to place 4 items in n^2 positions such that no row/column contains more than one

Given an integer N where 4 ≤ N ≤ 100. There are N lines vertically and N lines horizontally. So, There are N2 intersections. The task is find the number of ways to place 4 items in these N2 positions such that each row and column contain not more than one item.

Examples:

Input: N = 4

Output: 24

Input: N = 5
Output: 600

Approach: The number of ways to choose 4 horizontal lines that will have items from n is nC4. There are n ways to place an item on the first of these lines. Given the place of the first item, there are n – 1 ways to place an item on the second of these lines because one of the vertical lines is already taken. Given the places of the first and second items, there are n – 2 ways to place an item on the third line and the same way n – 3 for the fourth item. The total number of ways to place items on selected 4 horizontal paths is n * (n – 1) * (n – 2) * (n – 3). So the result is nC4 * n * (n – 1) * (n – 2) * (n – 3).

Below is the implementation of the above approach:

C++

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// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
  
// Function to return the number of ways
// to place 4 items in n^2 positions
long long NumberofWays(int n)
{
    long long x = (1LL * (n) * (n - 1) * (n - 2) * (n - 3))
                  / (4 * 3 * 2 * 1);
    long long y = (1LL * (n) * (n - 1) * (n - 2) * (n - 3));
  
    return (1LL * x * y);
}
  
// Driver code
int main()
{
    int n = 4;
    cout << NumberofWays(n);
  
    return 0;
}

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Java

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// Java implementation of the approach
class GFG
{
  
// Function to return the number of ways 
// to place 4 items in n^2 positions 
static long NumberofWays(int n) 
    long x = (1l * (n) * (n - 1) * (n - 2) * (n - 3)) 
                / (4 * 3 * 2 * 1); 
    long y = (1l * (n) * (n - 1) * (n - 2) * (n - 3)); 
  
    return (1l * x * y); 
  
// Driver code 
public static void main(String args[])
    int n = 4
    System.out.println( NumberofWays(n)); 
}
  
// This code is contributed by Arnab Kundu

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Python3

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# python implementation of the approach 
  
# Function to return the number of ways 
# to place 4 items in n^2 positions 
def NumbersofWays(n):
    x = (n * (n - 1) * (n - 2) * (n - 3)) // (4 * 3 * 2 * 1)
    y = n * (n - 1) * (n - 2) * (n - 3)
  
    return x * y
  
# Driver code
n = 4
print(NumbersofWays(n))
  
# This code is contributed by Shrikant13

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C#

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// C# implementation of the approach 
using System;
  
class GFG
{
  
// Function to return the number of ways 
// to place 4 items in n^2 positions 
public static long NumberofWays(int n)
{
    long x = (1l * (n) * (n - 1) * (n - 2) * 
               (n - 3)) / (4 * 3 * 2 * 1);
    long y = (1l * (n) * (n - 1) * (n - 2) * 
                (n - 3));
  
    return (1l * x * y);
}
  
// Driver code 
public static void Main(string[] args)
{
    int n = 4;
    Console.WriteLine(NumberofWays(n));
}
}
  
// This code is contributed by Shrikant13

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PHP

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<?php
// PHP implementation of the approach
  
// Function to return the number of ways
// to place 4 items in n^2 positions
function NumberofWays($n)
{
    $x = (1 * ($n) * ($n - 1) *
                     ($n - 2) * ($n - 3)) /
                          (4 * 3 * 2 * 1);
    $y = (1 * ($n) * ($n - 1) * 
                     ($n - 2) * ($n - 3));
  
    return (1 * $x * $y);
}
  
// Driver code
$n = 4;
echo NumberofWays($n);
  
// This code is contributed by mits
?>

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Output:

24


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