Is there a way to calculate these two quantities if I'm throwing two twelve sided die and adding the results that's easier than going through all possible combination and their probabilities?
Thanks.
Look at this webpage
Scroll down to the fully expanded form of the expression.
That gives you all the impossibles.
For example the term $\displaystyle 11x^{14}$ tells us that there eleven ways to roll a sum of fourteen. So the probability is $\displaystyle \frac{11}{144}$ of getting a sum of fourteen.