<p>There are <code>n</code> people and <code>40</code> types of hats labeled from <code>1</code> to <code>40</code>.</p>

<p>Given a 2D integer array <code>hats</code>, where <code>hats[i]</code> is a list of all hats preferred by the <code>i<sup>th</sup></code> person.</p>

<p>Return <em>the number of ways that the <code>n</code> people wear different hats to each other</em>.</p>

<p>Since the answer may be too large, return it modulo <code>10<sup>9</sup> + 7</code>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> hats = [[3,4],[4,5],[5]]
<strong>Output:</strong> 1
<strong>Explanation:</strong> There is only one way to choose hats given the conditions. 
First person choose hat 3, Second person choose hat 4 and last one hat 5.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> hats = [[3,5,1],[3,5]]
<strong>Output:</strong> 4
<strong>Explanation:</strong> There are 4 ways to choose hats:
(3,5), (5,3), (1,3) and (1,5)
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> hats = [[1,2,3,4],[1,2,3,4],[1,2,3,4],[1,2,3,4]]
<strong>Output:</strong> 24
<strong>Explanation:</strong> Each person can choose hats labeled from 1 to 4.
Number of Permutations of (1,2,3,4) = 24.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>n == hats.length</code></li>
	<li><code>1 &lt;= n &lt;= 10</code></li>
	<li><code>1 &lt;= hats[i].length &lt;= 40</code></li>
	<li><code>1 &lt;= hats[i][j] &lt;= 40</code></li>
	<li><code>hats[i]</code> contains a list of <strong>unique</strong> integers.</li>
</ul>

class Solution 
{
public:
    int numberWays(vector<vector<int>>& hats) 
    {
        int n = hats.size();
        std::unordered_map<int, std::vector<int>> map;
        for (int i = 0; i < n; ++i)
        {
            for (auto hat : hats[i])
                map[hat].push_back(i);
        }

        int mod = 1e9 + 7;
        std::memset(dp_, 0, sizeof(dp_));
        
        dp_[0][0] = 1;
        for (int i = 1; i <= 40; ++i)
        {
            for (int state = 0; state < (1 << n); ++state)
            {
                dp_[i][state] = dp_[i - 1][state];
                for (auto person : map[i])
                {
                    if (((state >> person) & 0b01) == 0)
                        continue;

                    dp_[i][state] = (dp_[i][state] + dp_[i - 1][state ^ (1 << person)]) % mod;
                }
            }
        }

        return dp_[40][(1 << n) - 1];
    }

private:
    int dp_[45][(1 << 10) + 1];
};
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