<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 MOD = 1e9+7;
    int goFind(int hat, int mask, int &n, unordered_map<int, vector<int>> &mp, vector<vector<int>> &dp) {
        if(hat == 0) {
            return __builtin_popcount(mask) == n;
        }
        if(dp[hat][mask] != -1) {
            return dp[hat][mask];
        }
        int not_take = goFind(hat-1, mask, n, mp, dp)%MOD;
        int take = 0;
        for(auto &it:mp[hat]) {
            if(((mask>>it)&1) == 0) {
                take = (take + goFind(hat-1, ((mask)|(1<<it)), n, mp, dp))%MOD;
            }
        }
        return dp[hat][mask] = (take + not_take)%MOD;
    }
    int numberWays(vector<vector<int>>& hats) {
        unordered_map<int, vector<int>> mp;
        int n = hats.size();
        vector<vector<int>> dp(41, vector<int> ((1<<n)+1, -1));
        for(int i=0;i<hats.size();i++) {
            for(int j=0;j<hats[i].size();j++) {
                mp[hats[i][j]].push_back(i);
            }
        }
        return goFind(40, 0, n, mp, dp);
    }
};
14400