<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 {
private:
    const int mod = int(1e9) + 7;
public:

    int dphelper(int hat, int personsAssigned, const int& allAssigned, vector<vector<int>>& dp, const vector<vector<int>>& hat2people) {
        const int n_hats = hat2people.size();
        if(personsAssigned == allAssigned) return 1;
        if(hat == n_hats) return 0;

        if(dp[hat][personsAssigned] != -1) return dp[hat][personsAssigned];

        // Skip hat
        int res = dphelper(hat+1, personsAssigned, allAssigned, dp, hat2people);

        for(const int& person : hat2people[hat]) {
            if(personsAssigned & (1 << person)) continue;
            res = (res + dphelper(hat + 1, personsAssigned | (1 << person), allAssigned, dp, hat2people)) % mod;
        }
        
        return dp[hat][personsAssigned] = res;
    }


    int numberWays(vector<vector<int>>& hats) {
        const int n = hats.size();
        const int n_hats = 41;

        vector<vector<int>> hat2people(n_hats, vector<int>());

        for(int person = 0; person < n; person++) {
            for(const int& hat : hats[person]) {
                hat2people[hat].push_back(person);
            }
        }

        vector<vector<int>> dp(n_hats, vector<int>(1024, -1));

        return dphelper(1, 0, (1 << n) - 1, dp, hat2people);

    }
};
25600