
Random Permutation
Suppose $\displaystyle \pi $ is a random permutation of $\displaystyle \mathbb{Z}_n $. How would you determine the probability that $\displaystyle \pi(i+1)\pi(i) \pmod n < n/2 $? We know that $\displaystyle \pi(i+1)\pi(i) $ is maximum at $\displaystyle n1 $ in $\displaystyle \mathbb{Z}_n $.

Perhaps we could use InclusionExclusion? In other words, find the probability that $\displaystyle \pi(i+1)\pi(i) > n/2 $ (i.e. the $\displaystyle i $th and $\displaystyle i+1 $st places differ by more than $\displaystyle n/2 $?