<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:
    
      int numberWays(vector<vector<int>>& hats) {
    int n = hats.size();  // Number of people
    int mod = 1e9 + 7;
    vector<vector<int>> peopleWithHat(41);  // Hats are 1-indexed up to 40
    
    // Mapping hats to people
    for (int i = 0; i < n; ++i) {
        for (int hat : hats[i]) {
            peopleWithHat[hat].push_back(i);
        }
    }

    vector<int> dp(1 << n, 0);
    dp[0] = 1;

    // Try each hat
    for (int h = 1; h <= 40; ++h) {
        vector<int> next_dp = dp;
        for (int person : peopleWithHat[h]) {
            for (int mask = 0; mask < (1 << n); ++mask) {
                if (!(mask & (1 << person))) {  // If person is not yet wearing a hat
                    int next_mask = mask | (1 << person);
                    next_dp[next_mask] = (next_dp[next_mask] + dp[mask]) % mod;
                }
            }
        }
        dp = move(next_dp);
    }

    return dp[(1 << n) - 1];
}  
    
   
};
12400