<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:
    def is_valid(self, mask: int, people: int) -> bool:
        if (mask & (1<<people)) != 0:
            return False
        else:
            return True
        
    def update(self, mask: int, people: int) -> int:
        next_mask = (mask | (1<<people))
        return next_mask
    
    def dfs(self, idx: int, mask: int) -> int:
        if idx == len(self.options):
            if mask == self.terminal:
                return 1
            else:
                return 0
        elif (idx, mask) in self.cache:
            return self.cache[(idx, mask)]
        else:
            cur = self.options[idx]
            # 1. not select any people to wear current hat
            res = self.dfs(idx+1, mask)
            # 2. let different person who doesn't have a hat to wear current hat
            for people in self.rev_hat[cur]:
                if self.is_valid(mask, people):
                    next_mask = self.update(mask, people)
                    tmp = self.dfs(idx+1, next_mask)
                    res = (res+tmp)%self.mod
            self.cache[(idx, mask)] = res
            return res
                
    
    def numberWays(self, hats: List[List[int]]) -> int:
        self.rev_hat = {}
        self.options = []
        self.n = len(hats)
        self.cache = {}
        self.terminal = 2**self.n-1
        self.mod = 10**9+7
        for i in range(len(hats)):
            for j in range(len(hats[i])):
                if hats[i][j] not in self.rev_hat:
                    self.rev_hat[hats[i][j]] = set()
                self.rev_hat[hats[i][j]].add(i)
        self.options = list(self.rev_hat.keys())
        res = self.dfs(0,0)
        return res
                
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