Solve in the set of non-negative integers.
It seems that would work but I'm not totally sure. Is there any "official" solution or is guesswork the only way to go?
I believe this should be in the Number Theory Section.
You might want to consider the following expression which is equivalent :
You can factorize this, then divide the RHS to the number-only LHS factor. Then try to see when it actually is an integer ()
EDIT : this might not give all solutions. You can also study the equation which is conceptually simpler.
If x=y, then we'd have 2*x!=z!
But unless x=1, it's not possible to have a solution.
So (1,1,2) is a solution.
If x<y (wlog), we can divide each side by x! and we'd have
It's obvious that y<z. So the RHS can be written
Since all of these are integers, we must have and
Again, since all the factors are integers, we must have
Thus there are only 2 solutions : (0,1,2) and (1,1,2)