<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:
    const int MOD = 1e9 + 7;
    
    // Memoization table to store intermediate results
    vector<vector<int>> dp;
    
    // Dynamic programming function to calculate the number of ways
    int dfs(int hat, int mask, vector<vector<int>>& hats_to_people, int done) {
        if (mask == done) {
            return 1; // All people are assigned hats
        }
        
        if (hat > 40) {
            return 0; // No more hats to assign
        }
        
        if (dp[hat][mask] != -1) {
            return dp[hat][mask]; // Return cached result
        }
        
        int ans = dfs(hat + 1, mask, hats_to_people, done); // Skip current hat
        
        // Try assigning current hat to each person who can wear it
        for (int person : hats_to_people[hat]) {
            if ((mask & (1 << person)) == 0) { // Check if person hasn't got a hat yet
                ans = (ans + dfs(hat + 1, mask | (1 << person), hats_to_people, done)) % MOD;
            }
        }
        
        return dp[hat][mask] = ans; // Cache and return the result
    }
    
    int numberWays(vector<vector<int>>& hats) {
        // hats_to_people maps each hat to a list of people who can wear it
        vector<vector<int>> hats_to_people(41); // Hats are numbered from 1 to 40
        
        int n = hats.size();
        for (int i = 0; i < n; ++i) {
            for (int hat : hats[i]) {
                hats_to_people[hat].push_back(i);
            }
        }
        
        int done = (1 << n) - 1; // All people are assigned when the mask has all bits set to 1
        dp = vector<vector<int>>(41, vector<int>(1 << n, -1)); // Initialize dp table with -1
        
        return dfs(1, 0, hats_to_people, done); // Start with hat 1 and no person assigned a hat
    }
};

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