@professionalbronco Yes, that definitely works! It's interesting how both strategies are useful in different cases- Prof. Loh's method is faster when considering larger numbers, but yours is quicker in this case with smaller numbers 🙂 Nice!

If we are trying to subtract the number of lines which aren't triangles for a question with a square of bigger dimensions (say a \(20 \times 20 \) square), wouldn't it be more tedious to count the number of lines rather than a \(3 \times 3 \) square? Is there a shortcut to count these "bad" 3 dot combinations?

In the question it says that Alicia has to be first in line, so there are \(3!=6\) ways, which is equal to the ways they can be arranged in a circle. The question only asks you if the answers are the same.