<p>There is a donuts shop that bakes donuts in batches of <code>batchSize</code>. They have a rule where they must serve <strong>all</strong> of the donuts of a batch before serving any donuts of the next batch. You are given an integer <code>batchSize</code> and an integer array <code>groups</code>, where <code>groups[i]</code> denotes that there is a group of <code>groups[i]</code> customers that will visit the shop. Each customer will get exactly one donut.</p>

<p>When a group visits the shop, all customers of the group must be served before serving any of the following groups. A group will be happy if they all get fresh donuts. That is, the first customer of the group does not receive a donut that was left over from the previous group.</p>

<p>You can freely rearrange the ordering of the groups. Return <em>the <strong>maximum</strong> possible number of happy groups after rearranging the groups.</em></p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> batchSize = 3, groups = [1,2,3,4,5,6]
<strong>Output:</strong> 4
<strong>Explanation:</strong> You can arrange the groups as [6,2,4,5,1,3]. Then the 1<sup>st</sup>, 2<sup>nd</sup>, 4<sup>th</sup>, and 6<sup>th</sup> groups will be happy.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> batchSize = 4, groups = [1,3,2,5,2,2,1,6]
<strong>Output:</strong> 4
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= batchSize &lt;= 9</code></li>
	<li><code>1 &lt;= groups.length &lt;= 30</code></li>
	<li><code>1 &lt;= groups[i] &lt;= 10<sup>9</sup></code></li>
</ul>

from collections import Counter, defaultdict
from functools import lru_cache

class Solution:
    def maxHappyGroups(self, batchSize: int, groups: List[int]) -> int:
        # Count the number of groups with the same size
        count = Counter([g % batchSize for g in groups])
        
        # Initial state: no groups served yet, and the remainder is 0
        initial_state = tuple([count[i] for i in range(batchSize)]) + (0,)
        
        @lru_cache(None)
        def dp(state):
            # Base case: if all groups have been served, return 0
            if sum(state[:-1]) == 0:
                return 0
            
            # Calculate the maximum number of happy groups for the current state
            max_happy = 0
            remainder = state[-1]
            for group_size, group_count in enumerate(state[:-1]):
                if group_count > 0:
                    # Serve the current group
                    new_state = list(state)
                    new_state[group_size] -= 1
                    new_state[-1] = (remainder + group_size) % batchSize
                    new_state = tuple(new_state)
                    
                    # Recursively find the maximum number of happy groups
                    happy_groups = dp(new_state)
                    if remainder == 0:
                        happy_groups += 1
                    
                    max_happy = max(max_happy, happy_groups)
            
            return max_happy
        
        # Call the DP function with the initial state
        return dp(initial_state)

# Example usage:
solution = Solution()
batchSize = 3
groups = [1,2,3,4,5,6]
print(solution.maxHappyGroups(batchSize, groups))  # Output: 4
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