<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 {
    const int MOD = 1e9+7;
    int n;

int helper (int curr, int mask, vector <vector <int>> &arr, vector <vector <int>> &dp){
    if (mask == (1 << n) - 1 ){
        return 1;
    }

    if (curr == 40) return 0;

    if (dp[curr][mask] != -1) return dp[curr][mask];

    int cnt = 0;
    for (int a : arr[curr]){
        if ((mask >> a) & 1) continue;
        cnt = (helper (curr+1, mask | (1 << a), arr, dp) + cnt) % MOD; // wear 
    }
    cnt = (helper (curr+1, mask, arr, dp) + cnt) % MOD; // do not wear
    return dp[curr][mask] = cnt;
}

public:
    int numberWays(vector<vector<int>>& hats) {
        
        n = hats.size();
        vector <vector <int>> arr (40);

        for (int i = 0; i < n; i++){
            for (int h : hats[i]){
                arr[h-1].push_back(i);
            }
        }
        vector <vector <int>> dp (41, vector <int> (1 << 10, -1));
        return helper (0, 0, arr, dp);
    }
};
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