We consider, first the following problems.

1. What is the maximum number of intersection points determined by n lines in the same plane?

2. What is the maximum number of lines determined by n points? Does is matter that the points are in the same plane?

3. What is the maximum number of planes determined by n points is space?

 

Now consider this problem

4. What is the maximum of regions space can be divided into with 5 planes? n planes?

Hint: Consider first:

What is the maximum number of regions a plane can be divided into with 5 lines? n lines?

What is the maximum number of regions a line can be divided into with 5 points? n points?