However, adding that 1<a<b<c, also allows trivial solutions: which? What condition should be added to disallow such solutions?

Finally, adding that condition, can you find any solution to the problem?

There are an infinite number of trivial solutions. If we allow b to equal a! - 1, then c is equal to a!. For example, if a=3 and b=5, then since a!=6 and c!=5!*6=b!*a!, the solution is trivial. These trivial solutions get incalculable very quickly.

I don't believe there are any quick solutions if you eliminate that condition.