If five distinct lines lie in a plane, then at most how many distinct points can lie on two or more of these lines?
6
7
8
9
10.
Each line can only cut 4 other lines.
For 5 lines, there will be a maximum of 5*4/2 = 10 intersections. The product has to be divided by two because each intersection had been counted twice, once for each line.
To figure out the maximum number of distinct points that can lie on two or more of these lines, we can take a systematic approach.
First, let's consider the maximum number of distinct points that can lie on one line. For a line in a plane, there are infinitely many points, so the maximum number of distinct points on one line is infinity.
Now, let's think about the number of intersections between two lines. When two distinct lines intersect in a plane, they do so at a single point. So each pair of lines can have at most one distinct point in common.
To find the number of intersections between five distinct lines, we can use combinations. We choose 2 out of 5 lines to form a pair. The number of ways to choose 2 items out of 5 is given by the combination formula:
C(5, 2) = 5! / (2!(5 - 2)!), which simplifies to 10.
So there can be at most 10 distinct points where the lines intersect in pairs.
However, we also need to count the distinct points where three or more lines intersect at the same point. Let's consider the worst-case scenario where all five lines intersect at a single point. In this case, we have only one extra distinct point.
Adding the number of intersections between pairs and the extra distinct point where all five lines intersect, we get a total of 10 + 1 = 11 distinct points.
Therefore, the correct answer is not among the options provided. The maximum number of distinct points that can lie on two or more of the five lines is 11.