@versatilemole i could give as many solutions as i can to this problem, and then people can add on solutions and count how many ways we can make 100.

Not really 🙂 Notice how Elmo has three blocks. It wouldn't be $$\binom{9}{3}$$ because you use a separate block every time so you would have to choose the first block, then the second, and so on. Try to see where you can go from there 😉

Note: Sorry this is so late I doubt you're active on this forum anymore.

@knowledgeabledog Thanks!

I am, he kinda cracked at minecraft

@sereneseagull Huh weird I got the same answer and I believe it's a typo...

@Bella Nice! 😄 That's definitely a valid way of doing it-- splitting into cases based on whether the triangles are equilateral, isosceles, or scalene (no two sides with the same lengths) is a good choice, because then it makes it harder to accidentally miss a triangle. Just goes to show that there are often multiple ways to solve a problem!

Does this mean that we can flip the WB block around, and make it to become only BW instead, with the combination
\(BWBWBW\), such that it ends in W?

@Kaiser hello, Im desolate101 looks like you really like Warrior Cats? Me too!

@geniusraccoon Hmmm I think its because the three gears need to be in a triangle to move, because the gears wont work one vs one on a line, you will be missing the first and the third if u use a line

@Bella Both are actually correct. I believe Professor Loh first found 27 (mod 4) to be 3, and then doubled this to get 6. Professor Loh is emphasizing the simplification step, which makes working with larger numbers easier. And if we simplify this further, 6 is 2 (mod 4). So your calculator is not wrong; 27 + 27 is 2 (mod 4). But noting Professor Loh's strategy will help you solve these types of problems more efficiently 🙂