How do you prove these answers mathematically? I understand what the correct answers are but I don't know how to get there with out plugging and chugging.

7. Suppose that you have n dice, each a different color, all unbiased and 6-sided. (a) If you roll them all at once, how many distinguishable outcomes are there? (b) Given two distinguishable dice, (say one is red and one is white) what is the most probable sum of their face values on a given throw of the pair? (That is, which sum between two and twelve has the greatest number of different ways of occurring?) (c) What is the probability of observing the most probable sum