Hi. Im a freshman student.

This question taken from the book by Matoussek and Nestril.

The statement goes:

Can someone please hint me as to how to adress this problem. Answers would be appreciated too but use spoiler tags or different font color so that I may read it after a few attempts by myself.Let us draw n lines in the plane in such a way that no two are parallel and no three intersect in a common point. Prove that the plane is divided into exactly n(n+1)/2 +1 parts by the lines

The only conclusion I have drawn is that the addition of an extra line to a plane containing n lines splits (n+1) regions into two parts.

For example :

# One line on a plane:

Two parts

# Add the (n+1)-th line ( i.e, now there are two lines):

n+1 planes split (i.e, 1+1=2, 2 planes split to form total 4 planes)

#so on...

now what?

Havnt really reached this part.Similarly, consider n planes in the 3-dimensional space in general position (no two are parallel, any three have exactly one point in common, and no four have a common point). What is the number of regions into which these planes partition the space?