Counting Problem

There are $N$ bags. Each bag contains $M$ balls numbered $1$ to $M$. Note that for every $i \in [1, M]$, there is exactly one ball with number $i$ in each bag. You took exactly one ball from each bag and arranged the balls in a row, with the $i$-th ball being from the $i$-th bag. Given a number $K$, count the number of arrangements such that: - There exists **at least** one ball numbered $K$ in