Ten distinct lines lie in the same plane. No pair of these lines is parallel to each other, and no more than two of the lines intersect at any one point. How many points lie on more than one of these ten lines? A)10 B)36 C)45 D)55 E)100

This is so similar to a question I was recently asked on the Tumblr Q&A that it must be from the same source. When the question asks “how many points lie on more than one of these ten lines,” it’s just asking for the number of intersections. So follow the same logic I used to solve the other question:

- Draw the first line. It’s all by itself, so it intersects nothing.
- Draw the second line. It intersects the first line. That’s 1 intersection.
- Draw the third line. It intersects both existing lines. That’s 2 more intersections.
- Draw the fourth line. It intersects all three existing lines. That’s 3 more intersections…

…and just keep going. The first four lines give you 0 + 1 + 2 + 3 intersections. The next six will continue that pattern. The number of intersections of those ten lines will be:

0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45

*I’m now realizing that I’ve done this question before, too. Check out my other solution (same approach, different formatting) here.*