<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:
        happy = 0
        freq = [0] * batchSize

        for g in groups:
            g %= batchSize
            if g == 0:
                happy += 1
            elif freq[batchSize - g] > 0:
                freq[batchSize - g] -= 1
                happy += 1
            else:
                freq[g] += 1

        return happy + self.dp(tuple(freq), 0, batchSize)

    def __init__(self):
        self.mem = {}

    def dp(self, freq: tuple[int], remainder: int, batchSize: int) -> int:
        hashed = self.hash(freq)
        if hashed in self.mem:
            return self.mem[hashed]

        ans = 0

        if any(f != 0 for f in freq):
            for i in range(len(freq)):
                if freq[i] > 0:
                    new_freq = list(freq)
                    new_freq[i] -= 1
                    ans = max(ans, self.dp(tuple(new_freq), (remainder + i) % batchSize, batchSize))
            if remainder == 0:
                ans += 1

        self.mem[hashed] = ans
        return ans

    def hash(self, freq: tuple[int]) -> str:
        return "#" + "#".join(map(str, freq))

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