Math 300

Groupwork

Combination Problems

Dr. Wilson

 

5. In problem 2, assume that there is no point inside the circle where three of the line meet. How many points of intersection are there inside the circle?

 

 

The picture from problem 2,

 

 

will not work in this problem, because there are several point where more than two lines intersect, and this will throw off our theoretical count. We will need to adjust the points somewhat so that there is no point inside the circle where more than two lines intersect.

 

 

 

We see in the figure above that it is possible to arrange 8 points on a circle so that there is no point inside the circle where more than two of the lines meet. We also see that there are quite a few points of intersection inside the circle for these lines. While it is possible to count them, it would probably be better to solve enough of the simpler problems to be able to find a pattern which would work in general and enable us to solve this problem.

 

If there are 1, 2, or 3 points on the circle

 

 

there are no points of intersection inside the circle. The first time we actually get a point of intersection inside the circle is with 4 points.

 

 

 

It gets a little more interesting with 5

 

 

There are 5 points of intersection.

 

With 6

 

 

 

 

there are 15 points of intersection. With 6 points, we need to take a little bit of care to make sure that there is no point where more than two lines meet.

 

At this point, the figures are getting more and more complicated, so let's see if we can deduce the pattern at this point.

 

 

We can find these numbers in Pascal's triangle

 

 

These are the number of 4 element subsets of a given set. The next number in the sequence is 35 which should be the number of points of intersection inside the circle if there are 7 points on the circle.

 

 

you can count that there are 35 points of intersection inside, and in our problem, there are 70 points inside the circle.

The reason that this procedure works is because if you have a point inside a circle where 2 lines intersect,

 

 

each line is determined by 2 points on the circle, so if you have two lines, they will be determined by 4 points on the circle. This sets up a one to one correspondence between points inside the circle and 4 element subsets of the set of points on the circle.

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Groupwork