Number of rectangles in N*M grid

13 Jun, 2022
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
 
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 rectanglesIf 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)/2After 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)/4So 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++
Java
Python3
C#
PHP
Javascript
// 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 Code to count number of// rectangles in N*M gridimport 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 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 coden, m = 5, 4print(rectCount(n, m)) # This code is contributed by Anant Agarwal.
// C# Code to count number of// rectangles in N*M gridusing 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 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?>
<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)
This article is contributed by Pranav. If you like GeeksforGeeks and would like to contribute, you can also write an article using write.geeksforgeeks.org or mail your article to review-team@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above. 
jit_t
arora_hritik
perfecto_pride
decode2207
hasani
square-rectangle
Geometric
Mathematical
Mathematical
Geometric
Writing code in comment?
Please use ide.geeksforgeeks.org,
generate link and share the link here.
Program for distance between two points on earth
Find if two rectangles overlap
Check whether triangle is valid or not if sides are given
Line Clipping | Set 1 (Cohen–Sutherland Algorithm)
Program for Point of Intersection of Two Lines
Program for Fibonacci numbers
Set in C++ Standard Template Library (STL)
Write a program to print all permutations of a given string
C++ Data Types
Merge two sorted arrays