<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 numberWays(self, hats: List[List[int]]) -> int:
        MOD = 10**9 + 7
        n = len(hats)
        max_hat = 40
        
        # Create a list of lists where hat_to_people[i] contains the list of people who prefer hat i
        hat_to_people = [[] for _ in range(max_hat + 1)]
        for person, hat_list in enumerate(hats):
            for hat in hat_list:
                hat_to_people[hat].append(person)
        
        # dp[i][j] represents the number of ways to assign hats to people represented by bitmask j
        dp = [[0] * (1 << n) for _ in range(max_hat + 1)]
        dp[0][0] = 1  # Base case: 0 hats and 0 people
        
        for hat in range(1, max_hat + 1):
            for mask in range(1 << n):
                dp[hat][mask] = (dp[hat][mask] + dp[hat - 1][mask]) % MOD  # Not choosing this hat
                for person in hat_to_people[hat]:
                    if mask & (1 << person):
                        dp[hat][mask] = (dp[hat][mask] + dp[hat - 1][mask ^ (1 << person)]) % MOD
        
        return dp[max_hat][(1 << n) - 1]

# Example usage:
solution = Solution()
hats = [[3,4],[4,5],[5]]
print(solution.numberWays(hats))  # Output: 1

hats = [[3,5,1],[3,5]]
print(solution.numberWays(hats))  # Output: 4

hats = [[1,2,3,4],[1,2,3,4],[1,2,3,4],[1,2,3,4]]
print(solution.numberWays(hats))  # Output: 24
17900