If we take a random permutation of a sequence of $2k$ elements, $X_1, X_2, \ldots X_k, X_{k+1}, \ldots, X_{2k}$. What's the probability that $X_1, X_2, .. X_k$ and $X_{k+1}, \ldots, X_{2k}$ both keep their relative orders in the new sequence?
My guess is, these 2 event are independent because whether one happen doesn't change the other's probability. So we have
$ \Pr \{ [\text {both sub-sequence keep relative order}]\} = (\frac 1 {n!})^2 $
What do you think about it? Is it possible to get the result by counting how many permutations fulfill the condition?