I know from Number of ways of choosing at least $k$ objects out of $n$ and Partial sum of rows of Pascal's triangle that given $n$ objects the number of ways to choose at least $k$ has no closed form solution. But is there a way to find a non trivial($2^n$ is a trivial upper bound) upper bound on the same?
Edit:
Found a similar question on Math Overflow that also answered my question. Here's the link : https://mathoverflow.net/a/17236
