Let x1, x2,..., x20 be distinct points on the x-axis and let y1, y2,...,y13 be distinct points on the y-axis. For each pair xi, yi, draw the segment connecting xi yo y1. Assume that no three of these segments intersect in one point. How many points of intersection are there? (Don't count intersection points on either of the axes) If you can explain why... it will be awesome cuz i'm trying the best i can.
To solve this problem, let's start by considering the possible points of intersection.
We have 20 distinct points on the x-axis and 13 distinct points on the y-axis. Let's label the x-axis points as x1, x2, ..., x20 from left to right, and the y-axis points as y1, y2, ..., y13 from bottom to top.
Now, let's consider a single segment connecting one xi with y1. This segment intersects the y-axis at the point (xi, y1).
When we draw all the segments connecting each xi with y1, we actually create a set of 20 points (xi, y1) that lie on the y-axis.
Similarly, for each yi on the y-axis, we draw a segment connecting it with each xi on the x-axis. This creates a set of 13 points (x1, yi) that lie on the x-axis.
Now, we need to find the number of intersection points between these two sets of points.
Since each point in the first set lies on the y-axis and each point in the second set lies on the x-axis, the intersection points will lie inside the rectangular region formed by the sets.
The number of intersection points will be equal to the product of the number of points in each set because each point in the first set can only intersect with one point in the second set, and vice versa.
Therefore, the number of intersection points is 20 * 13 = 260.
So, there are a total of 260 intersection points in this scenario.
Note: We have assumed that no three segments intersect at a single point, as mentioned in the question.