<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 typing import List
from collections import Counter
from functools import lru_cache


class Solution:
    def maxHappyGroups(self, batchSize: int, groups: List[int]) -> int:
        # Counter for remainders when dividing group sizes by batchSize
        remainder_count = Counter(group % batchSize for group in groups)
        
        # Count the zero remainders (they are always happy groups)
        happy_groups = remainder_count[0]
        del remainder_count[0]
        
        # Cache to memoize the dfs function to avoid redundant calculations
        @lru_cache(None)
        def dfs(remaining):
            # Current total "leftover" donuts in the batch
            current_total = sum((i * count) for i, count in enumerate(remaining))
            # Default happy groups counts when continuing from here
            max_happy = 0
            
            for i in range(1, batchSize):
                if remaining[i] > 0:
                    new_remaining = list(remaining)
                    new_remaining[i] -= 1
                    new_remaining = tuple(new_remaining)
                    
                    # Calculate if the current batch makes the next group happy
                    happy = dfs(new_remaining)
                    if (current_total - i) % batchSize == 0:
                        happy += 1
                    
                    max_happy = max(max_happy, happy)
            
            return max_happy
        
        # Convert remainder_count to a tuple to facilitate caching
        remaining = tuple(remainder_count.get(i, 0) for i in range(batchSize))
        
        return happy_groups + dfs(remaining)
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