That’s true. But I thought doing it by hand would require writing a tediously large table because you have 6^3 possible roll results to give the probability of, if you were actually going to write the “likelihood of various dice rolls”. I suppose the appropriate compromise is a symbolic manipulation program like Mathematica, which can work with exact numbers easily while automating the creation of the table. (If anyone can explain the problem, it would be great if they could link to a document demonstrating the solution on somewhere like http://www.mathics.net/ .) Or is there an easier, simpler way to solve this by hand?

There are 6^4 possible rolls. But you don't need to list them all. You can take shortcuts.

After doing it by hand, I became interested in how I could do it by computer... :D