Count of rectangles possible from N and M straight lines parallel to X and Y axis respectively

Given two integers N and M, where N straight lines are parallel to the X-axis and M straight lines are parallel to Y-axis, the task is to calculate the number of rectangles that can be formed by these lines.

Examples:

Input: N = 3, M = 6 
Output: 45 
Explanation:
There are total 45 rectangles possible with 3 lines parallel to x axis and 6 lines parallel to y axis.
Input: N = 2, M = 4 
Output: 6 
Explanation:
There are total 6 rectangles possible with 2 lines parallel to x axis and 4 lines parallel to y axis. 

Approach:
To solve the problem mentioned above we need to observe that a rectangle is formed by 4 straight lines in which opposite sides are parallel and the angle between any two sides is 90. Hence, for every rectangle, two sides need to be parallel to X-axis and the other two sides need to be parallel to Y-axis. 

• Number of ways to select two lines parallel to X axis = ^NC₂ and the Number of ways to select two lines parallel to Y axis = ^MC₂ .
• So the total number of rectangles  = ^NC_{2 *} ^MC₂  = [ N * (N – 1) / 2 ] * [ M * (M – 1) / 2 ]

Below is implementation of above approach: 

C++

// C++ Program to count number of
// rectangles formed by N lines
// parallel to X axis M lines
// parallel to Y axis
#include <bits/stdc++.h>
using namespace std;

// Function to calculate
// number of rectangles
int count_rectangles(int N, int M)
{
    // Total number of ways to
    // select two lines
    // parallel to X axis
    int p_x = (N * (N - 1)) / 2;

    // Total number of ways
    // to select two lines
    // parallel to Y axis
    int p_y = (M * (M - 1)) / 2;

    // Total number of rectangles
    return p_x * p_y;
}

// Driver Program
int main()
{

    int N = 3;

    int M = 6;

    cout << count_rectangles(N, M);
}

Java

// Java Program to count number of
// rectangles formed by N lines
// parallel to X axis M lines
// parallel to Y axis
class GFG{

// Function to calculate
// number of rectangles
static int count_rectangles(int N, int M)
{
    // Total number of ways to
    // select two lines
    // parallel to X axis
    int p_x = (N * (N - 1)) / 2;

    // Total number of ways
    // to select two lines
    // parallel to Y axis
    int p_y = (M * (M - 1)) / 2;

    // Total number of rectangles
    return p_x * p_y;
}

// Driver Program
public static void main(String[] args)
{
    int N = 3;
    int M = 6;

    System.out.print(count_rectangles(N, M));
}
}

// This code is contributed by sapnasingh4991

Python3

# Python3 program to count number of rectangles
# formed by N lines parallel to X axis
# and M lines parallel to Y axis
def count_rectangles(N, M):

    # Total number of ways to select
    # two lines parallel to X axis
    p_x = (N * (N - 1)) // 2

    # Total number of ways to select
    # two lines parallel to Y axis
    p_y = (M * (M - 1)) // 2

    # Total number of rectangles
    return p_x * p_y

# Driver code
N = 3
M = 6

print(count_rectangles(N, M))

# This code is contributed by himanshu77

C#

// C# Program to count number of
// rectangles formed by N lines
// parallel to X axis M lines
// parallel to Y axis
using System;
class GFG{

// Function to calculate
// number of rectangles
static int count_rectangles(int N, int M)
{
    // Total number of ways to
    // select two lines
    // parallel to X axis
    int p_x = (N * (N - 1)) / 2;

    // Total number of ways
    // to select two lines
    // parallel to Y axis
    int p_y = (M * (M - 1)) / 2;

    // Total number of rectangles
    return p_x * p_y;
}

// Driver Program
public static void Main()
{
    int N = 3;
    int M = 6;

    Console.Write(count_rectangles(N, M));
}
}

// This code is contributed by Code_mech

Javascript

<script>

// JavaScript Program to count number of
// rectangles formed by N lines
// parallel to X axis M lines
// parallel to Y axis

// Function to calculate
// number of rectangles
function count_rectangles(N, M)
{
    // Total number of ways to
    // select two lines
    // parallel to X axis
    let p_x = (N * (N - 1)) / 2;

    // Total number of ways
    // to select two lines
    // parallel to Y axis
    let p_y = (M * (M - 1)) / 2;

    // Total number of rectangles
    return p_x * p_y;
}

// Driver Code

    let N = 3;
    let M = 6;

    document.write(count_rectangles(N, M));

</script>

Output:

45

Time Complexity: O(1), as constant operations are being performed.
Auxiliary Space: O(1), as constant space is being used.