How do you find the number of states unlabeled particles can be in a box split in half?

Question

I came across this problem doing research into entropy and I was wanting to simulate diffusion with a live count of the entropy of the system. In the entropy formula, you use "Ω", which is the number of possible microstates of a system (I think). Anyways I found a video explaining this where they showed each possible setup of a system and the microstates it has and I am wondering what the equation is to find this out. In this image one of the possible states of the system is 5 particles on one half of the box and 5 on the other. This system has 252 possible microstates. I am wondering what the equation is to get this 252 number from the state of the system.

Answer 1

252 is "ten choose 5", the number of ways to choose five particles from 10. Think of the particles as being labeled 1,2,3...10 and think about the ways you can put all of them on the left-hand side. There's one way to do that, of course. What about to put one on the right-hand side? There's ten, one for each particle. For two particles it becomes 10 choose 2, or 10*9/2 because you can choose 10 for the first particle and only nine for the second, but order doesn't matter, so you would double count choosing 1,2 and 2,1 as separate microstates if you didn't divide by two. In general,

$$n \textrm{ choose } m = \frac{n!}{m!(n-m)!}.$$

Answer 2

The answer is $${10\choose{5}}=\frac{10!}{5!(10-5)!}$$ Look up for example https://www.calculatorsoup.com/calculators/discretemathematics/combinations.php