Combinatorics – Lines Cutting Regions in Geometry

combinatoricseuclidean-geometrygeometry

$15$ lines are drawn in a plane such that $4$ of them are parallel.

  1. What is the maximum number of regions into which the plane is divided?

  2. How many of the regions have finite area (bounded)?

My Attempt

  1. The total number of regions that $n$ lines divides the plane into, is given by $\dfrac{n(n+1)}{2} +1$.

    So I got $\dfrac{15\cdot16}{2} + 1 = 121$. But $4$ lines are parallel. So instead of getting $\dfrac{4\cdot5}{2} + 1 = 11$ regions, we only get $4 + 1 = 5$ regions. So the actual number of regions are $121 – 11 + 5 = 115$

  2. A single line divides the plane in $2$ infinite (unbounded) regions.

    So $15$ lines can give $30$ infinite regions. But $4$ of them are parallel. So we get only $5$ infinite regions instead of $8$. So the actual number of infinite regions are $30 – 8 + 5 = 27$. Hence, the total finite (bounded) regions are $115-27 = 88$.

While i got the answer to the second question as $88$ the original answer is $85$. Can someone please tell me where I am going wrong?

Also if there is any better approach please let me know.

Best Answer

OK. So for b, you should subtract 30, and this is why:

Imagine drawing 4 parallel lines. You have 5 unbounded regions. Add a line that is not parallel to them. You get 10 unbounded regions. Now add another line that is not parallel to any of the existing lines. You will see that although it creates a certain number of bounded regions, it will also add 2 infinite regions to the already existing 10. And this will happen to every line from here on. At that point we have 6 lines and 12 infinite regions. Now since every other line will add 2 more, we get that if we have 15 lines, we have at least 30 unbounded regions.

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