Each two of ten points in the plane intersect and no three lines pass through the same point. In how many regions are the lines dividing the plane?
The number of regions created by the intersection of lines in a plane can be calculated using a formula known as Euler's formula. Euler's formula states that for a planar graph, the number of regions (including the infinite region) is equal to the number of edges minus the number of vertices plus 1.
In this case, we have 10 lines in the plane. The number of intersections can be calculated as the sum of the first 9 positive integers since each line intersects with every other line exactly once. The sum of the first 9 positive integers is (9 * 10) / 2 = 45.
So, the number of vertices is 45, and the number of edges is the number of lines, which is 10.
Using Euler's formula, the number of regions is given by:
Number of regions = Number of edges - Number of vertices + 1 = 10 - 45 + 1 = -34 + 1 = -33
However, it doesn't make sense to have a negative number of regions, so in this case, we should consider the value of 1 for the infinite region. Therefore, the number of regions created by the intersection of the lines in the plane is 1 -(- 33) = 34.
