Say you have two 6 sided dice numbered from 1 to 6. How would you calculate the probability of rolling 2, 3, 4, ..., 12?
My professor told me the generating function $(x+x^2+x^3+x^4+x^5+x^6)^2$ is incorrect for starting the problem because the dice are treated as distinct. That I should be looking for the answer using identical dice and "sum". How would this be solved?
Well if your dice are identical then you can simply say that for finding the probability of rolling a certain sum $s$ we need to find the number of ways $x_1 + x_2$ can be equal to $s$ or rather we need to find the number of integral solutions to the equation $$x_1 + x_2 = s$$ where $$x_1,x_2 \equiv 1,2,3,4,5,6$$ We can solve this rather easily and then we can use the argument for probability.