Let $A=\{1,2,...,10\}$ and Let $\mathcal A = \{ T \subset A | |T|=4\}$.Consider an arbitrary function $f:\mathcal A \longrightarrow A$. Prove that there exists a subset $S$ of $A$ with five elements such that $f(S-\{r\}) \neq r$ for each $r \in S$
My idea is to use the fact $\binom{10}{4}$ is strictly smaller than $\binom{10}{5}$
Any hint is welcome to solve this. Thanks in advance