<s>[INST] There are $N$ doors numbered from $1$ to $N$. There are $K$ persons who can open each door. Each person can open at most one door, but all $K$ persons are identical and cannot be distinguished. We call a subset of the persons a "group". Here, "group" means that it is possible for the doors to be unlocked by this group of people in any order. For example, if there are three doors and three persons, person $A$ unlocks door 1, person $B$ unlocks the door 2, person $C$ unlocks the door 3. In this case, $\{A\}$, $\{B\}$, $\{C\}$, $\{A, B\}$, $\{A, C\}$, $\{B, C\}$, $\{A, B, C\}$ are the groups that can open the doors. We want to know the maximum number of groups that can open all the doors. Here, in order to calculate the maximum number of groups, we need to find the optimal strategy to lock the doors. Now, let's consider how many groups can be formed when there are $i$ doors and $j$ persons. Let's denote this number as $f(i, j)$. It can be seen that $f(0, j) = 1$, $f(i, 0) = 0$. Also, $f(1, 1) = 2$, $f(2, 2) = 5$, $f(2, 3) = 8$, $f(3, 3) = 14$. From this, we can propose the following recurrent relation:

$$f(i, j) = \sum_{k=1}^{j} \binom{j}{k} f(i - 1, j - k).$$

From this formula and the previously mentioned values, we can calculate the maximum number of groups for $i$ doors and $j$ persons. 

Now, we are interested in the following problem: what is the maximum number of groups that can open all the doors, assuming that we have $N$ doors and $K$ persons with $N \leq K$?[/INST]