If you roll $n$ six-sided dice what is the probability that the sum of the numbers you roll is multiple of $k?$
I was wondering which combinatoric strategies would be useful to solve problems of these types. I often just find the probability for each multiple of $k$ and sum them up. Is there a more efficient way to approach this question?
Let $$ \begin{align} f(x) &=x+x^2+x^3+x^4+x^5+x^6\\ &=\frac{x-x^7}{1-x}\tag{1} \end{align} $$ then $f(x)^n$ is the generating function of the number of ways to roll a given number on $n$ dice.
Thus, since $$ \frac1k\sum_{j=0}^{k-1}e^{2\pi i\,jm/k}=[k\mid m]\tag{2} $$ where $[\dots]$ are Iverson Brackets, the probability of rolling $m\pmod{k}$ on $n$ dice is $$ \frac1{k6^n}\sum_{j=0}^{k-1}e^{-2\pi i\,jm/k}f\!\left(e^{2\pi i\,j/k}\right)^n\tag{3} $$ For this question, we can set $m=0$ in $(3)$.