I have a sequence of length $N$ created by removing some elements (possibly zero) from a permutation of numbers $(1,2,\dots,N)$. When an element is removed, the length of the sequence doesn't change, but there is an empty spot left where the removed element was, denoted by $\ast$. You also have an integer $K$.
Let's call a permutation $p_1,p_2,\dots,p_N$ good if:
it is possible replace empty spots in the permutation by numbers in such a way that we obtain the permutation $p$ where the number of positions $i$ ( $1 < i \leq N$ ) such that $p_i > p_{i−1}$ is equal to $K$ My task is to find the number of good permutations.
Eg. When $n=3$ elements and $K=1$: Given the array $(2 ,* ,*)$, then the answer$=2$ explanation: two possible ways $(2,3,1)$ and $(2,1,3)$
I have tried as $f=$number of * in array, ans as $(f)!-k$
but this gives me wrong answer.
pls help me derive the formula