<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:
        for i in range(len(groups)):
            groups[i] = groups[i]%batchSize

        cache = defaultdict(int)

        groups.sort()
        ans = 0

        for i in groups:
            if i == 0:
                ans += 1
                groups.remove(i)

        if groups:
            ans += 1

        n = len(groups)

        dp = [0 for i in range(n+1)]

        def dfs(curr_tot, active_sum,rem,batch_size,n):

            if not rem:
                return curr_tot

            if active_sum % batch_size == 0 and len(rem) != n:
                curr_tot += 1
                active_sum = 0

            if dp[n-len(rem)]>curr_tot:
                return 0
            else:
                dp[n-len(rem)] = curr_tot

            if (tuple(rem) in cache):
                return cache[tuple(rem)] + curr_tot

            ret = 0

            for i in range(len(rem)):

                a = rem.pop(i)
                ret = max(ret,dfs(curr_tot,active_sum + a,rem,batch_size,n))
                rem.insert(i,a)

            cache[tuple(rem)] = ret-curr_tot

            return ret

        return dfs(0,0,groups,batchSize,n) + ans
                
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