I'm not sure where the boundary is between using "algebraic formulas" and not. I would solve this using combinations (it is 6C3, because you are choosing 3 of the 6 places to put a black tile), and that is normally done using something that could be called a formula: 6!/(3!3!) = (6*5*4)/(3*2*1) = 20. There are ways to count without that formula, but I'd consider them harder. The best way I can think of for someone who hasn't learned combinations is what is called an "orderly list": write out each arrangement in such a way as to be sure to cover all possibilities. If done well, that really is not too slow; but I probably wouldn't write a test that demanded that.

Then again, I'm surprised that this would be expected by any method at those ages. Do you know what they have been taught in this area, that they might be expected to use? Have they said anything about what thoughts the problem evokes?

I checked out what the ICAS is, and it looks like it may include an occasional challenge like this. Can you show us where you got this sample problem, and what level it is for? Where do they mention avoiding formulas? Any further information might help us see what might be expected.