Let $a$ and $n$ be positive integers with $a \leqslant n$. Is there a transitive action of $S_n$ on a set $A$ of $\binom{n}{a}$ elements?
My attempt was to fix a set $\{1, \cdots, n\}$ and consider the set $C$ of all choices $\{i_1, \cdots, i_a\}$.
However, if I just let $S_n$ act by $\sigma(\{i_1, \cdots, i_a\}) = \{\sigma(i_1), \cdots, \sigma(i_a)\}$, it will not be a group action as it is not closed.
I could not however think about any other actions.
I moved on trying to prove there is no such an action but get stuck.
Any help would really be appreciated.