Sum of products over k-permutations
When $K = N$, the polynomial you are trying to compute is known as the permanent. The best algorithms for computing the permanent use $\tilde{O}(2^N)$ arithmetic operations, see Wikipedia. The VP≠VNP conjecture implies that the permanent cannot be computed using a polynomial number of operations.
For general $K$, your function is a known extension of the permanent. Ryser's algorithm, one of the $\tilde{O}(2^N)$ algorithms for the square case, extends to the rectangular case as well.