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:
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  = NC2 * MC = [ N * (N – 1) / 2 ] * [ M * (M – 1) / 2 ]

Below is implementation of above approach: 
 



C++

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// 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);
}

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Java

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// 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

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Python3

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# 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

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C#

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// 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

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Output:

45

Time Complexity: O(1)

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