I like the way you are thinking! You are going deep into this question, and that is absolutely wonderful. The good thing is that this idea, the number of ways to choose 2 things out of 6 things, (generally, to choose x things out of y things) is a very fundamental concept. If you can understand this, you can do a TON of advanced problems!
Prof. Loh solved two separate questions: 1.) How many intersection points are there? and 2.) How many triangles can you make?
For the first question, let's color the lines like this:
The 6 × 5 comes from the fact that there are 6 groups of ways, with 5 ways in each group. I'll illustrate two of the groups right here:
Group 1 (Ways start with Red)
Group 2: (Ways start with Green)
Group 3: (Ways start with Yellow) (not pictured)
Group 4: (Ways start with Blue) (not pictured)
Group 5: (Ways start with Black) (not pictured)
Group 6: (Ways start with Dark Green) (not pictured)
This is an easy way to count the ways, because without drawing the ways out, we can see that the 6 comes from the number of lines, and the 5 came from the number of other lines left over after we chose the first one! However, the downside to this easy way is that we have to consider the double-counted ways. In the Red Group and Green Group, there is a way that occurs in both: the intersection of the red line and the green line.
Actually, every way has a copy, and if you drew out all the lines for all the groups, you would see this!
So we divide 2 to the number of ways that we drew out, which gives
as the number of different intersection points. (Note that this was just a warm-up question; it was the answer to the hint. This is to show you the idea of how to choose x things from y things, and isn't needed for finding the number of triangles.)
The second question is to find the number of triangles.
I like to think of the top as meaning that there are 6 large groups, each with 5 small groups, each with 4 ways in them.
I'll only draw one of the largest groups, the one with first line red:
Yes, all those ways are in the first group (ways starting with red): 5 × 4 ways! The common characteristic between all these groups is that they have a red line. The picture above contains five sub-groups; you'll notice that in each group, there is always one other color that is present (for example, the last group includes ways with red and blue lines).
There are five other large groups (a group for ways which all contain the yellow line, a group for ways which all contain the black line, a group for ways which all contain the blue line, a group for ways which all contain the light green line, and a group for ways which all contain the dark green line). Each of those other five large groups also each contains five subgroups, each of which contain four ways.
However, we have counted some ways more than once. Each of the ways gets counted six times, as shown in the picture below. The number "1" means that this color is the large group color, the number "2" means that this color is the subgroup color, etc.
By symmetry, since each way is counted six times, we can divide the total ways we got earlier by 6 (which is 3!). So the number of triangles is f8844604-b38d-4616-8e89-230fea77f24b-image.png
I hope this helps! We love helping our students through their struggles, so please don't hesitate to ask more questions. I am more than happy to answer them!
The Daily Challenge Team