Maximum Intersection Points of Circles: A Step-by-Step Calculation

The question asks for the maximum number of points of intersection possible when we have 5 non-overlapping circles. To find the maximum number of intersection points among multiple geometric shapes, we generally consider the case where every possible pair of shapes intersects at the maximum possible number of points for a pair.

Understanding Pairwise Intersection of Circles

Let's first consider two distinct circles. What is the maximum number of points at which two circles can intersect?

• Two circles can touch at exactly one point (tangent).
• Two circles can intersect at exactly two distinct points.
• Two circles can intersect at zero points (if they are separate or one is inside the other, but the question specifies non-overlapping, implying they don't contain each other).

For the maximum number of intersection points for a given number of circles, we assume that every pair of circles intersects at the maximum possible number of points. For two distinct circles, this maximum is 2 points.

Calculating the Number of Pairs of Circles

We have 5 circles. To find the total maximum number of intersection points, we need to consider every possible pair of circles. The number of ways to choose 2 circles out of 5 is given by the combination formula:

The number of combinations of selecting \(k\) items from a set of \(n\) items is denoted by \(\binom{n}{k}\) and calculated as:

\(\binom{n}{k} = \frac{n!}{k!(n-k)!}\)

In this problem, \(n=5\) (the total number of circles) and \(k=2\) (the number of circles in each pair). So, the number of pairs of circles is:

\(\binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5!}{2!3!}\)

Let's expand the factorials:

\(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\)

\(2! = 2 \times 1 = 2\)

\(3! = 3 \times 2 \times 1 = 6\)

Substitute these values back into the combination formula:

\(\binom{5}{2} = \frac{120}{(2)(6)} = \frac{120}{12} = 10\)

So, there are 10 distinct pairs of circles among the 5 circles.

Calculating the Maximum Total Intersection Points

Since each pair of circles can intersect at a maximum of 2 points, and there are 10 pairs of circles, the maximum total number of intersection points is the product of the number of pairs and the maximum intersection points per pair.

Maximum total intersection points = (Number of pairs of circles) \(\times\) (Maximum intersection points per pair)

Maximum total intersection points = \(10 \times 2 = 20\)

It is possible to arrange 5 non-overlapping circles such that every pair intersects at 2 distinct points, resulting in a total of 20 intersection points. This arrangement ensures that no three circles intersect at the same point, maximizing the total count.

Therefore, the maximum number of points of intersection of 5 non-overlapping circles is 20.

Revision Table: Maximum Intersection Points

Additional Information: Maximum Intersection Points of Other Shapes

The method of finding the maximum intersection points often involves calculating the number of pairs and multiplying by the maximum pairwise intersections. Let's look at other shapes:

• Lines: Maximum intersection points of \(n\) distinct lines is \(\binom{n}{2}\). This is because two lines intersect at most at 1 point, and there are \(\binom{n}{2}\) pairs of lines. Total = \(\binom{n}{2} \times 1\).
• Polygons: The maximum intersection points of two convex n-gons can be up to \(2n\). The maximum for \(k\) convex n-gons is more complex.
• Ellipses: Similar to circles, two distinct ellipses can intersect at a maximum of 4 points. The maximum intersection points of \(n\) ellipses would be \(\binom{n}{2} \times 4\).

For 5 circles, the maximum intersection points calculation is a direct application of counting pairs and considering maximum pairwise intersections, assuming a general position where no three circles intersect at a single point and every pair intersects twice.