EDIT:Let $S$ be any finite set and suppose $x \notin S$. Let $K=S \cup \{x\}$.
1.Prove that $P(K)$ is the disjoint union of $P(S)$ and $X=\{T \subset K : x \in T\}$. That is, show that $P(K) = P(S) \cup X$.
2.Prove that every element of $X$ is the union of a subset of $S$ with $\{x\}$, and that if you take different subsets of $S$ you get different elements of $X$. Argue that, therefore, $X$ has the same number of elements as $P(S)$.
3.Argue that the previous two parts allow you to conclude that if $S$ is a finite set, then $P(K)$ has twice as many elements as $P(S)$.