I derived a general result about the sum of numbers on a die whose correctness I was wondering about, so could someone please confirm it? It is as follows:
If $k$ dice are thrown, and the sum we desire is $n$, we are looking for positive integer solutions to $$x_1 + x_2 + \cdots + x_k = n$$ with each $x_i$ satisfying $1 \leq x_i \leq 6$. This can be found by calculating the positive integer solutions without the restrictions, which will be $\binom{n-1}{k-1}$. If $n \geq 6 + k$, we will need to subtract the number of positive solutions to $$(y_1 + 6) + (y_2 + 6) + \cdots + (y_k + 6) = n$$ $$y_1 + y_2 + \cdots + y_k = n - 6k$$ which will be $\binom{n-6k-1}{k-1}$.
Therefore the final answer is $$\binom{n-1}{k-1} - \binom{n-6k-1}{k-1}$$