Number of rectangles in N*M grid

We are given a N*M grid, print the number of rectangles in it.
Examples:

Input  : N = 2, M = 2
Output : 9
There are 4 rectangles of size 1 x 1.
There are 2 rectangles of size 1 x 2
There are 2 rectangles of size 2 x 1
There is one rectangle of size 2 x 2.

Input  : N = 5, M = 4
Output : 150

Input :  N = 4, M = 3
Output: 60

Brute Force Approach :

• Iterate over all possible pairs of horizontal lines.
• Iterate over all possible pairs of vertical lines.
• count the number of rectangles that can be formed using these lines. 

Below is the code for the above approach :

C++

#include <bits/stdc++.h>
using namespace std;

int rectCount(int n, int m)
{
    int count = 0;
    for(int i=1; i<=n; i++) // iterating over all possible pairs of horizontal lines
    {
        for(int j=1; j<=m; j++) // iterating over all possible pairs of vertical lines
        {
            count += (n-i+1)*(m-j+1); // counting the number of rectangles that can be formed using these lines
        }
    }
    return count;
}

/* driver code */
int main()
{
    int n = 5, m = 4;
    cout << rectCount(n, m);
    return 0;
}

Output

150

Time Complexity : O(N^2)
Space Complexity : O(1)

We have discussed counting number of squares in a n x m grid,
Let us derive a formula for number of rectangles.
If the grid is 1×1, there is 1 rectangle. 
If the grid is 2×1, there will be 2 + 1 = 3 rectangles 
If it grid is 3×1, there will be 3 + 2 + 1 = 6 rectangles. 
we can say that for N*1 there will be N + (N-1) + (n-2) … + 1 = (N)(N+1)/2 rectangles
If we add one more column to N×1, firstly we will have as many rectangles in the 2nd column as the first, 
and then we have that same number of 2×M rectangles. 
So N×2 = 3 (N)(N+1)/2
After deducing this we can say 
For N*M we’ll have (M)(M+1)/2 (N)(N+1)/2 = M(M+1)(N)(N+1)/4
So the formula for total rectangles will be M(M+1)(N)(N+1)/4 

.

Combinatorial Logic:

N*M grid can be represented as (N+1) vertical lines and (M+1) horizontal lines.
In a rectangle, we need two distinct horizontal and two distinct verticals.
So going by the logic of Combinatorial Mathematics we can choose 2 vertical lines and 2 horizontal lines to form a rectangle. And total number of these combinations is the number of rectangles possible in the grid.

Total Number of Rectangles in N*M grid: ^N+1C₂ * ^M+1C₂ = (N*(N+1)/2!)*(M*(M+1)/2!) = N*(N+1)*M*(M+1)/4

C++

// C++ program to count number of rectangles
// in a n x m grid
#include <bits/stdc++.h>
using namespace std;

int rectCount(int n, int m)
{
    return (m * n * (n + 1) * (m + 1)) / 4;
}

/* driver code */
int main()
{
    int n = 5, m = 4;
    cout << rectCount(n, m);
    return 0;
}

Java

// JAVA Code to count number of
// rectangles in N*M grid
import java.util.*;

class GFG {

    public static long  rectCount(int n, int m)
    {
        return (m * n * (n + 1) * (m + 1)) / 4;
    }

    /* Driver program to test above function */
    public static void main(String[] args)
    {
        int n = 5, m = 4;
       System.out.println(rectCount(n, m));
    }
}

// This code is contributed by Arnav Kr. Mandal.

Python3

# Python3 program to count number
# of rectangles in a n x m grid

def rectCount(n, m):

    return (m * n * (n + 1) * (m + 1)) // 4

# Driver code
n, m = 5, 4
print(rectCount(n, m))

# This code is contributed by Anant Agarwal.

C#

// C# Code to count number of
// rectangles in N*M grid
using System;

class GFG {

    public static long  rectCount(int n, int m)
    {
        return (m * n * (n + 1) * (m + 1)) / 4;
    }

    // Driver program
    public static void Main()
    {
        int n = 5, m = 4;
       Console.WriteLine(rectCount(n, m));
    }
}

// This code is contributed by Anant Agarwal.

PHP

<?php
// PHP program to count
// number of rectangles
// in a n x m grid

function rectCount($n, $m)
{
    return ($m * $n *
           ($n + 1) *
           ($m + 1)) / 4;
}

// Driver Code
$n = 5;
$m = 4;
echo rectCount($n, $m);

// This code is contributed
// by ajit
?>

Javascript

<script>

    // Javascript Code to count number
    // of rectangles in N*M grid

    function rectCount(n, m)
    {
        return parseInt((m * n * (n + 1) *
                        (m + 1)) / 4, 10);
    }

      let n = 5, m = 4;
      document.write(rectCount(n, m));

</script>

Output

150

Time complexity: O(1)
Auxiliary Space: O(1), since no extra space has been taken.

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