<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>

#include <vector>
#include <unordered_map>

using namespace std;

class Solution {
public:
    int numberWays(vector<vector<int>>& hats) {
        const int MOD = 1e9 + 7;
        int n = hats.size();
        
        // Create a mapping of hats to people
        vector<vector<int>> hat_to_people(41); // hat_to_people[hat] = {people who prefer this hat}
        for (int person = 0; person < n; ++person) {
            for (int hat : hats[person]) {
                hat_to_people[hat].push_back(person);
            }
        }
        
        // dp[mask] stores the number of ways to assign hats given people in the mask have hats
        vector<int> dp(1 << n, 0);
        dp[0] = 1; // Base case: 1 way to assign hats when no one has a hat
        
        // Iterate over all hats
        for (int hat = 1; hat <= 40; ++hat) {
            vector<int> new_dp = dp; // We will update dp for the current hat
            
            for (int mask = 0; mask < (1 << n); ++mask) {
                if (dp[mask] == 0) continue; // No way for this mask
                
                // Try to assign the current hat to each person who prefers it
                for (int person : hat_to_people[hat]) {
                    if ((mask & (1 << person)) == 0) { // If this person hasn't been assigned a hat
                        int new_mask = mask | (1 << person);
                        new_dp[new_mask] = (new_dp[new_mask] + dp[mask]) % MOD;
                    }
                }
            }
            
            dp = new_dp; // Update dp with new_dp after considering this hat
        }
        
        // The final answer is the number of ways to assign hats to all people
        return dp[(1 << n) - 1];
    }
};

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