Given 4, 6-sided dice, how many different outcomes (e.g. three 6s, one 4) are there if order doesn't matter?

The answer, which I believe to be correct, is 126. I obtained this by systematically listing the possibilities. Obviously, such a procedure would be unmanageable with larger sets of dice with more sides. I tried to use permutations, combinations, and the multiplication principle, but I'm still having trouble. I also tried to write the systematic listing process in sigma notation and then convert it into a polynomial equation, but also failed at that. Any ideas?