<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:
    const long MOD = 1e9 + 7;
    
    long count(int hat, unordered_map<int, vector<int>>& p, int people, vector<vector<long>>& dp) {
        if (people == dp.size()-1) return 1;
        if (hat > 40) return 0;
        if (dp[people][hat] != -1) return dp[people][hat];
        
        long res = count(hat+1, p, people, dp);
        
        for (auto person: p[hat]) if ((people & 1<<person) == 0) {
            int more = people | 1<<person;
            res = (res + count(hat+1, p, more, dp)) % MOD;
        }
        
        return dp[people][hat] = res;
    }
    
    int numberWays(vector<vector<int>>& hats) {
        int n = hats.size(); // number of people
        unordered_map<int, vector<int>> p;
        for (int person=0; person<n; person++) {
            for (auto& hat: hats[person]) {
                p[hat].push_back(person);
            }
        }
        vector<vector<long>> dp(1 << n, vector<long>(41, -1));
        int people = 0;
        long res = count(1, p, people, dp);
        return res;
    }
};
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