Maximum Number of Groups Getting Fresh Donuts
<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>
Bit Manipulation; Memoization; Array; Dynamic Programming; Bitmask
class Solution {
public:
    int maxHappyGroups(int batchSize, vector<int>& groups) {
        using ll = long long;
        unordered_map<ll, int> f;
        ll state = 0;
        int ans = 0;
        for (auto& v : groups) {
            int i = v % batchSize;
            ans += i == 0;
            if (i) {
                state += 1ll << (i * 5);
            }
        }
        function<int(ll, int)> dfs = [&](ll state, int mod) {
            if (f.count(state)) {
                return f[state];
            }
            int res = 0;
            int x = mod == 0;
            for (int i = 1; i < batchSize; ++i) {
                if (state >> (i * 5) & 31) {
                    int t = dfs(state - (1ll << (i * 5)), (mod + i) % batchSize);
                    res = max(res, t + x);
                }
            }
            return f[state] = res;
        };
        ans += dfs(state, 0);
        return ans;
    }
};