(a) \(X_1, X_2,\ldots,X_9\)are nine points on the circumference of circle \(O\). Line segments are drawn connecting each pair of points.

How many line segments are drawn?

(b) \(X_1, X_2,\ldots,X_9\) are nine points on the circumference of circle \(O\) . Line segments are drawn connecting each pair of points.

What is the largest number of different points inside the circle at which at least two of these line segments intersect? (Remember that the points are not necessarily evenly spaced around the circle.)

Mr.Owl
Oct 29, 2017

#1**+2 **

Here's the first one

From the 1st point, 8 segments can be drawn to the other points

From the 2nd point, 7 segments can br drawn to the other points [ a segment to this point has already been drawn from the 1st point ]

So.....continuing the pattern.....the number of total segments is just the sum of the 1st eight positive integers = [ 8 * 9 ] / 2 = 36

To see this a little more clearly, note that we are really counting all the different pairs we can form by choosing any 2 points from 9 = C(9,2) = 36

CPhill
Oct 29, 2017