<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>


// #define ll long long int 
// int mod = 1e9 + 7;


// class Solution {
//   long long countWays(int person, vector<vector<int>>& hats, vector<bool>& used) {
//         int n = hats.size();
//         if (person == n) {  
//             return 1;
//         }
         
    
//         long long ways = 0;
    
//         for (int hat : hats[person]) {
//             if (!used[hat]) {  
//                 used[hat] = true;  
//                 ways = (ways + countWays(person + 1, hats, used)) % mod;  
//                 used[hat] = false;
//             }
//         }
        
//         return ways;
//     }
// public:
//     int numberWays(vector<vector<int>>& hats) {
//      int n = hats.size(); 
//      vector<bool> used(n,false);
     
    
//         return countWays(0, hats, used);
        
//     }
// };

class Solution {
    int mod = 1e9 + 7; // To avoid overflow
    int dp[41][1 << 10]; // DP table: dp[hat][mask], where mask is a bitmask representing the hats already assigned
    vector<vector<int>> personHats; 
    int countWays(int hat, int mask, int n) {
        if (mask == (1 << n) - 1) { 
            return 1;
        }
        if (hat > 40) { 
            return 0;
        }
        if (dp[hat][mask] != -1) { 
            return dp[hat][mask];
        }
        int ways = countWays(hat + 1, mask, n) % mod;
        for (int person : personHats[hat]) {
            if (!(mask & (1 << person))) {
                ways = (ways + countWays(hat + 1, mask | (1 << person), n)) % mod;
            }
        }
        
        return dp[hat][mask] = ways;
    }

public:
    int numberWays(vector<vector<int>>& hats) {
        int n = hats.size(); // Number of people
        personHats = vector<vector<int>>(41); 
        for (int i = 0; i < n; i++) {
            for (int hat : hats[i]) {
                personHats[hat].push_back(i);
            }
        }
        
        memset(dp, -1, sizeof(dp));
        return countWays(1, 0, n);
    }
};
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