So I was reading this question: What are the chances that out of $n$ people playing rock, paper, scissors, only two choices are picked?
and saw this type of answer in other place as well. The:
But you can also represent this as a multinomial distribution with the PMF of the form:
$P(X=x,Y=y,Z=z)=\frac{n!}{x!y!z!}(\frac{1}{3})^n$, where n=x+y+z.
So my question is why. I want to break it into small "chunks" to understand it better. From where did the fraction of $\frac{n!}{x!y!z!}$ come.
I do think that I understand why I need to make a power of $n$ for $\frac{1}{3}$ , because the probability of getting any of the options (rock, scissors or paper) is $\frac{1}{3}$, and the probability is the same for all of them, so I just need to multiply it by the number of the total players which is $n$, right?
I think I just don't really understand how the joint PMF idea helps me or even works in general (?) I mean yes. I do know the whole theory about joint PMF, but I can't really understand how it worked here.
And also this whole multinomial theorem idea, is not very clear to me in this question and answer.
Thanks.