Are there any nice counting arguments that identify the sum
$$S_{n,k} = {n \choose k} - {n \choose k-1} + {n \choose k-2} + \cdots + (-1)^k {n \choose 0}$$
with a simpler, more conceptually-appealing expression?
This expression comes up as the type of rank of a homology group of a somewhat complicated object and I'm hopeful I might be inspired by a counting argument.
$$S_{n,k}=\binom{n-1}k.$$ Look at the subsets of $[n]=\{1,\ldots,n\}$ of size $\le k$, and pair off $A$ with $A\cup\{n\}$ for $A\subseteq[n-1]$. The unpaired sets are the $k$-element subsets of $[n-1]$.