<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>

const int mod = 1e9 + 7;
class Solution {
private:
	int fun(int &n, vector<int>*adj, vector<vector<int>>&dp, int s, int mask) {
		if (s == 41) {
			return (mask + 1 == (1 << n));
		}
		if (dp[mask][s] != -1)return dp[mask][s];
		int ans = fun(n, adj, dp, s + 1, mask);
		for (int i = 0; i < adj[s].size(); i++) {
			if ((mask & ((1 << adj[s][i]))) != (1 << adj[s][i])) {
				ans += fun(n, adj, dp, s + 1, mask | (1 << adj[s][i]));
				ans %= mod;
			}
		}
		return dp[mask][s] = ans;
	}
public:
	int numberWays(vector<vector<int>>& h) {
		int n = h.size();
		vector<int>adj[41];
		for (int i = 0; i < n; i++) {
			for (auto &j : h[i]) {
				adj[j].push_back(i);
			}
		}
		vector<vector<int>>dp((1 << h.size()), vector<int>(41, -1));
		return fun(n, adj, dp, 1, 0);
	}
};
13500