Let $X$ be the set $ \{ 1, 2, \cdots, 20 \} $ and let $P$ be the set of all 9-element subsets of $X$. Show that for any map $ f: P \mapsto X $ we can find a 10-element subset $Y$ of $X$, such that $ f(Y - \{ k \}) \ne k $ for any $k$ in $Y$.