<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 setbits(x) __builtin_popcount(x)
const int M = 1000000007;
class Solution {
    int solve(int mask, int num, vector<vector<bool>> &hat, vector<vector<int>> &dp, int n){
        if(setbits(mask)==n){
            return 1;
        }
        if(num==40) return 0;
        if(dp[mask][num]!=-1) return dp[mask][num];
        int ans = 0;
        ans += solve(mask, num+1, hat, dp, n);
        for(int i=0;i<n;i++){
            if(hat[i][num]==1 && !(mask&(1<<i))){
                ans += solve((mask|(1<<i)), num+1, hat, dp, n);
            }
            ans%=M;
        }
        return dp[mask][num] = ans%M;
    }
public:
    int numberWays(vector<vector<int>>& hats) {
        int n = hats.size();
        vector<vector<bool>> hat(n, vector<bool> (40,0));
        for(int i=0;i<n;i++){
            for(auto x:hats[i]){
                hat[i][x-1] = 1;
            }
        }
        vector<vector<int>> dp((1<<n), vector<int>(40,-1));
        return solve(0,0,hat,dp,n);
    }
};
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