For $N\in \mathbb{N}$, $M\in \mathbb{N}$, and $K\in \mathbb{N}$, $f(K)$ is given by \begin{equation} f(K) = \sum\limits_{i = 1}^{K} {\left( {\frac{{\left( {M - K} \right)!\left( {M - i} \right)!}}{{\left( {M - i - K} \right)!M!}}} \right)^{N - 1} \cdot \left( \begin{array}{c} K \\ i \\ \end{array} \right) \cdot \left( { - 1} \right)^{i - 1} } \end{equation}, where $M\ge K$.
We found that $f(K)$ is a convex function by plotting $f(K)$ for varying $K$. We would like to solve the above equation as a closed-form. Does anyone help us for solving it or suggesting any ideas?