<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:
    vector<vector<int>> dp;
    vector<vector<int>> htp;
    int mod = 1000000007;
    int full;
    int solve(int a, int b) {
        if (b >= full) return 1;
        if (a >= 40) return 0;
        if (dp[a][b] >= 0) return dp[a][b];

        long long res = solve(a + 1, b);
        for (auto i : htp[a]) {
            if (b & (1<<i)) continue;
            res = (res + solve(a + 1, b | (1<<i))) % mod;
        }
        return dp[a][b] = res;
    }

    int numberWays(vector<vector<int>>& hats) {
        htp = vector<vector<int>> (40, vector<int>());
        int n = hats.size();
        full = (1<<n) - 1;
        dp = vector<vector<int>>(40, vector<int>((1<<(n + 1)), -1));
        for (int i = 0; i < hats.size(); i++) {
            for (auto j : hats[i]) htp[j - 1].push_back(i);
        }

        return solve(0, 0);
    }
};
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