<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:
        n = len(hats)
        mod = 10**9 + 7
        max_hat = 40
        
        # Create a list of hat preferences for each person
        hat_prefs = [[] for _ in range(n)]
        for i in range(n):
            for hat in hats[i]:
                hat_prefs[i].append(hat - 1)  # Convert to 0-based index
        
        # dp[i][j] represents the number of ways to assign hats to people represented by bitmask i
        dp = [[0] * (1 << n) for _ in range(max_hat + 1)]
        dp[0][0] = 1  # Base case: no hats and no people
        
        # Iterate over each hat
        for i in range(1, max_hat + 1):
            for mask in range(1 << n):
                dp[i][mask] = (dp[i][mask] + dp[i-1][mask]) % mod  # Carry forward the previous state
                
                # Try to assign the current hat to each person
                for j in range(n):
                    if (mask & (1 << j)) == 0 and i - 1 in hat_prefs[j]:
                        dp[i][mask | (1 << j)] = (dp[i][mask | (1 << j)] + dp[i-1][mask]) % mod
        
        # The result is the number of ways to assign hats to all people
        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