#math problem solution

Word problem:

Imagine you have six potential heroes in a videogame, but you're only allowed to choose three of them per run. You choose all three before the first scene of the game, and the order in which you selected them determines how they appear in the story and how they relate to one another. Given that only three out of six can be present in the game each time you play, and that the order of the three chosen heroes does matter, how many variations of the plot of the game are possible to experience? (Presume that there are no other meaningful decisions in the game)

That’s a really cool question! If you don’t mind, I’m going to think out loud here.

My first though was to use information from a way to use a möbius strip. With möbius strips, it’s all about saying that (a,b) should be mapped onto (b,a). But with this problem, the opposite is true, since we want (a,b) and (b,a) to be their own things. The next issue that we come across is that we’re not looking at 2 values, but 3. That’s also easily fixable since 3 coordinates would just be mapped into 3D space, each coordinate point being a potential variation to the game. So, this problem could be visualized as the number of integer coordinate points in a cube 5 unit cubes tall.

From there, the problem is relatively simple. We can extrude each point to be a unit cube, making it a question of the volume of a new cube of 6 unit cubes tall. This is simply 6^3, or 216.

And there’s your answer! I’m not positive on it, but it feels pretty good so if anyone else wants to solve it I encourage it! I want to be proven wrong if I am so I can learn. I’m sure this problem could be solved with series or something like that, but sometimes the best solution in math is to do the thing that makes the most sense to you.