<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 dp[41][1<<10];
    int f(int i,int mask,int m,unordered_map<int,vector<int>>&hatToP,int n) {
        if (mask == (1<<n)-1) return 1;
        if (i==m+1) return 0;
        if (dp[i][mask] != -1) return dp[i][mask];
        if (!hatToP.count(i)) return f(i+1,mask,m,hatToP,n);
        // do not place hat i to any person
        int ans = f(i+1,mask,m,hatToP,n);
        vector<int>v = hatToP[i];
        for (auto p: v) {
            if (!(mask &(1<<p))) {
                int newmask = mask|(1<<p);
                ans =(ans+ f(i+1,newmask,m,hatToP,n)%mod)%mod;
            }
        } 
        return  dp[i][mask] = ans;
    }
    int numberWays(vector<vector<int>>& hats) {
        int n = hats.size();
        //int m = hats[0].size();
        unordered_map<int,vector<int>>hatToP;
        int maxH =0;
        for (int i=0;i<n;i++) {
            for (int j=0;j<hats[i].size();j++) {
                hatToP[hats[i][j]].push_back(i);
                maxH = max(maxH,hats[i][j]);
            }
        }
        //mask for people who have been paired with a hat.
        memset(dp,-1,sizeof(dp));
        return f(1,0,maxH,hatToP,n);
        
    }
};
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