Mathematics was born from an interest in counting.

Many problems in mathematics involve counting processes that often lead to extremely useful general formulas. For example, to count in how many ways we can combine n objects into groups of r such objects, we use the well-known formula:

Counting regions

This is an interesting combinatorial problem that can be solved in a simple way.

One hundred points are distributed over a circumference. The segment connecting any two of these points does not pass through the intersection point of two other segments. Into how many regions is the circle divided when all one hundred points are connected?

Let us first try to solve the problem with a smaller number of points. Examining cases with 2, 3, 4, and 5 points, we can observe that:

• 2 points generate \( 2 = 2^1 \) regions.

• 3 points generate \( 4 = 2^2 \) regions.

• 4 points generate \( 8 = 2^3 \) regions.

• 5 points generate \( 16 = 2^4 \) regions.

The results lead us to believe that 6 points would generate \( 2^5 = 32 \) regions, However, when we directly check what happens with 6 points, we see that 31 regions are generated, not 32. Therefore, the intended generalisation is not true.

Developing a formula that provides the number of regions obtained with 100 points

The segments connecting points in pairs will be called diagonals. Since for every two points we have a diagonal, their number is

and the number of intersection points of the diagonals is

since every 4 points determine two diagonals, which have one point in common.

Now we will describe a process that allows us to obtain the number of regions by successively eliminating diagonals.

When we remove one of the diagonals, the number of regions will decrease, since two regions that share a segment of the removed diagonal merge into a single region. Note that, in this process, the number of intersection points of each diagonal with those that have not yet been removed is considered. At the end, when all diagonals are successively removed, this number is equal to the total number of intersection points of all diagonals, that is,

Also, the number of regions decreases until it is reduced to a single region, when all the diagonals have been eliminated. We can then conclude that the number of regions eliminated is given by the total number of intersection points of all the diagonals, that is,

plus as many terms equal to 1 as there are diagonals. Thus,

Note: For n points, we have the same expression, simply replacing 100 with n.