How many ways can the US Supreme Court render a 5-4 decision?
In my opinion, a 5-4 can be 5 yes and 4 no, or 5 no and 4 yes, so I suppose that in the end, I would need to add 2 things together. Also, this does not vary much from the coin flipping problem, but I don't know where to start.
Would P(9,5) + (P,4) be the correct answer?
You need to define carefully what you mean by a different way. I would think the order the justices sign on to a decision does not matter, so there are $9 \choose 5$ ways to give $5$ yes and $4$ no. This would usually be $C(9,5)$, not $P(9,5)$. There are the same number for $5$ no and $4$ yes.
Sometimes the justices write several opinions, so there could be $2+2+1$ voting yes with three separate opinions. This would increase the number. The question should specify which is wanted.