Exactly. No need for 1000 simulations or other nonsense that people post to show off that they know some statistics. This has nothing to do with statistics, it is a pure probability problem. A simple enumeration, such as yours, is an analytically comprehensive proof, one that would be acknowledged as a proof by any mathematically literate person.
Except that when this problem was initially posed, it took simulations to get very prominent mathematicians to accept it.
Literal top of their field types.
At the time, it was hotly debated until simulations were done.
Now, it’s regarded as common knowledge.