Forming Quadrilaterals Using Points on a Circle

Suppose you have seven points marked on a circle. How many different quadrilaterals can be formed by selecting any 4 of these 7 points? This problem can be approached through the lens of combinatorics, which is a critical component in SEO as it helps in optimizing content to understand and answer specific user queries effectively.

Understanding the Problem

Given that the points are marked on a circle, we can utilize their polar coordinates, with the center of the circle as the origin (0,0) and radius r. Each point can be represented as (r, θi) where i 1, 2, 3, 4, 5, 6, 7, and θj i if j

Combinatorial Analysis

The key to solving this problem lies in understanding the combinatorial nature of the task. In combinatorics, the number of ways to choose r elements from a set of n elements is given by the binomial coefficient, often denoted as ( nC_r ) or ( _nC_r ), which is calculated as:

Formula: ( _nC_r frac{n!}{r!(n-r)!} )

For our specific problem, we need to find the number of ways to choose 4 points out of 7. This is mathematically represented as ( _7C_4 ).

Calculation

Let's perform the calculation step by step:

Step 1: Calculate ( 7! ) (Factorial of 7) ( 7! 7 times 6 times 5 times 4 times 3 times 2 times 1 5040 )

Step 2: Calculate ( 4! ) (Factorial of 4) ( 4! 4 times 3 times 2 times 1 24 )

Step 3: Calculate ( (7-4)! ) (Factorial of 3) ( 3! 3 times 2 times 1 6 )

Step 4: Apply the formula ( _7C_4 frac{7!}{4! times 3!} frac{5040}{24 times 6} frac{5040}{144} 35 )

Conclusion and Visualization

Thus, the number of different quadrilaterals that can be formed by selecting any 4 of the 7 points marked on the circle is 35. This can be visualized by considering the different configurations of the points. Here are some specific cases:

Case 1: First and last vertex as rθ1 (rθ1, rθ2, rθ3, rθ4, rθ1)
This is one of the possible configurations and can be counted. Case 2: First and last vertex as rθ2 (rθ2, rθ3, rθ4, rθ5, rθ2)
This configuration is also valid and contributes to the total number of quadrilaterals. Case 3: First and last vertex as rθ3 (rθ3, rθ4, rθ5, rθ6, rθ3)
This configuration is another valid choice. Case 4: First and last vertex as rθ4 (rθ4, rθ5, rθ6, rθ7, rθ4)
This is the final configuration and completes the count of 35 quadrilaterals.

This comprehensive approach not only solves the problem but also provides a clear understanding of the combinatorial principles at work. Understanding and applying such methods in your content can significantly enhance its SEO performance by addressing user queries effectively.

Keywords: combinatorics, circle points, quadrilaterals