Question 80
Let $S$ be a set of size $11$. A random $12$-tuple $(s_{1}, s_{2}, \ldots, s_{12})$ of elements of $S$ is chosen uniformly at random. Moreover, let $\pi: S \rightarrow S$ be a permutation of $S$ chosen uniformly at random. The probability that $s_{i+1} \neq \pi(s_{i})$ for all $1 \leq i \leq 12$ (where $s_{13}=s_{1}$) can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Compute $a$.