Past exam of the mathematics course of the University of Cambridge / 2025 / ia / Paper 3 / 5D / b

For $1\le k<n$, any nonidentity permutation moves some $k$-subset: choose a moved point and complete a subset so that it contains that point but not its image. Thus the kernel is trivial. For $k=n$ the action is trivial, so it is faithful only in the degenerate case $n=1$.