Let $n\in \mathbb{N}$ and $A_n=\{1,2,\dots,2n\}$. Calculate the cardinality of the following set $\{X\in P_k(A_n)|\forall r,s\in X.|r-s|\notin \{1,2\}\}$ for given $k\in \mathbb{N}$, $k\le n$.
Notes:
a. $P_k(A_n)$ means subsets of $A_n$ of size k.
b. The answer should be a function of $k$ and $n$.
I'm struggling with this one for a while so any direction would be appreciated.