Complexity of Subset Sum where the size of the subset is specified
There are $\binom{n}{k}$ $k$-subsets of an $n$ set, and $\binom{n}{k} = n (n - 1) \dotsm (n - k + 1) / k!$, which is $O(n^k)$, as you observe. The brute force complexity is a bit more (need to add up the numbers too and check), but that is ballpark.
This is polynomial for any fixed $k$, but not polynomial in $n$ if e.g. $k = n / 2$ (that is the Partition problem, known NP-complete).