<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>

class Solution:
    def maxHappyGroups(self, batchSize: int, groups: List[int]) -> int:
        remainder_count = defaultdict(int)
        happy_groups = 0
        
        for group in groups:
            remainder = group % batchSize
            if remainder == 0:
                happy_groups += 1
            else:
                remainder_count[remainder] += 1
        
        @lru_cache(None)
        def dp(remaining_batch, remainder_tuple):
            remainders = list(remainder_tuple)
            
            # If no groups left
            if sum(remainders) == 0:
                return 0
            
            # We try to serve a group that leaves `i` remainder
            max_happy = 0
            for i in range(1, batchSize):
                if remainders[i] > 0:
                    # Decrease the count for this remainder
                    new_remainders = remainders[:]
                    new_remainders[i] -= 1
                    
                    # Check if the current batch is full
                    if remaining_batch == 0:
                        # A new batch, this group will be happy
                        max_happy = max(max_happy, 1 + dp((batchSize - i) % batchSize, tuple(new_remainders)))
                    else:
                        # Continue with leftover donuts
                        max_happy = max(max_happy, dp((remaining_batch - i) % batchSize, tuple(new_remainders)))
            
            return max_happy

        happy_groups += dp(0, tuple(remainder_count[i] for i in range(batchSize)))
        return happy_groups
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