Calculate the maximum number of unique circles that can be formed by selecting 4 points from a set of 7 points in a plane. | Step-by-Step Solution

Understanding Maximum Circles from 7 Points

What We're Solving:

We have 7 points scattered on a plane, and we want to find the maximum number of unique circles we can draw, where each circle passes through exactly 4 of our points. The key insight is understanding what "concyclic points" means and when we get the maximum number of circles.

The Approach:

This is a combinatorics problem disguised as geometry. We need to consider:

• 1. How many ways can we choose 4 points from 7 points?
• 2. When do 4 points actually form a circle?
• 3. What arrangement gives us the maximum number of valid circles?

The beautiful part is that we're looking for the maximum possible, so we get to arrange our 7 points in the most favorable way possible!

Step-by-Step Solution:

Step 1: Calculate total combinations First, let's find how many ways we can select 4 points from 7 points:

• This is C(7,4) = 7!/(4! × 3!) = (7 × 6 × 5)/(3 × 2 × 1) = 35
• So there are 35 different groups of 4 points we could potentially use

Step 2: Understand the geometric constraint Any 4 points can form a circle UNLESS 3 or more of them are collinear (lie on the same straight line). When points are collinear, they can't all lie on the same circle!

Step 3: Find the optimal arrangement To maximize our circles, we want to minimize collinear points:

• If we place our 7 points so that NO THREE points are collinear, then every group of 4 points will be concyclic
• This is called "points in general position"
• In this optimal arrangement, we can use ALL 35 combinations!

Step 4: Verify this is achievable Can we actually place 7 points so no 3 are collinear? Absolutely! For example, place them randomly, or put them on a circle (though that's just one way).

The Answer:

The maximum number of unique circles is 35.

This occurs when the 7 points are arranged in "general position" (no three points collinear), allowing every possible combination of 4 points to form a valid circle.

Memory Tip:

Remember: "Maximum circles = maximum choices!" When we want the maximum number of circles, we arrange points to avoid the only thing that stops us - having 3+ points on a line. In the best case, every combination works, so our answer is simply C(7,4)!

Great job working through this geometric-combinatorics problem! The key insight is recognizing that geometry problems often have a combinatorics component, and optimization problems ask us to find the best-case scenario.