<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:
    const int MOD = 1e9 + 7 ;
    int f(vector<vector<int>>&a,int n,int mask,int i,vector<vector<int>>&dp){
        if(mask==(1<<n)-1) return 1;
        if(i>40) return 0;
        if(dp[mask][i]!=-1) return dp[mask][i];
        int ans=0;
        ans=f(a,n,mask,i+1,dp);
        for(int j=0;j<n;j++){
            if(a[j][i]==1&&((mask&(1<<j))==0)){
                ans=(ans+f(a,n,mask|(1<<j),i+1,dp))%MOD;
            }
        }
        return dp[mask][i]=ans;
    }
    int numberWays(vector<vector<int>>& hats) {
        int n=hats.size();
        vector<vector<int>>a(n+1,vector<int>(41,0));
        for(int i=0;i<n;i++){
            for(int j=0;j<hats[i].size();j++){
                int id=hats[i][j];
                a[i][id]=1;
            }
        }
        vector<vector<int>>dp(1025,vector<int>(41,-1));
        return f(a,n,0,0,dp);
    }
};
30400