From my years browsing math.SE, I've come to learn there's always some useful--albeit obscure--identity for every conceivable sum of combinatoric functions.
Mine is as simple as they come: $$S(n,m)=\sum _{k=0} ^m {n \choose k}$$ for $m \leq n$.
That is, a partial sum of binomial coefficients. Practically: the number of ways one can choose a subset of $m$ or fewer objects out of a set of $n$ objects.
Apologies if this is already answered elsewhere, but a search for "sum of binomial coefficients" turns up hundreds of permutations (no pun intended) of the question, none of which seem to be relevant to this specific case.
$${n\choose n-m}{\mbox{$_2$F$_1$}(1,-m;\,1+n-m;\,-1)}$$
where $\mbox{$_2$F$_1$}$ is a hypergeometric function.
See also OEIS sequence A008949.