<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:
    long long int solve(vector<vector<int>>& hats, vector<vector<long long int>>& dp, int ind, long long int mask, int people){
        //_______________________________________________________

        //T.C. : N*K*(2^K)

        //_______________________________________________________

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

        long long mod = 1e9+7;
        long long total = 0;

        for(int i=0;i<hats[ind].size();i++){
            if((mask & (1<<hats[ind][i])) ){
                total += solve(hats, dp, ind+1, mask ^ (1<<hats[ind][i]), people);
                total %= mod;
            }
        }

        dp[ind][mask] = total%mod;
        return dp[ind][mask];
    }
    long long int solve2(vector<vector<int>>& hats, vector<vector<long long int>>& dp, int hat, long long int mask, int people, vector<vector<int>>& data){
        //_______________________________________________________
        //T.C. : N*K*(2^K)
        //_______________________________________________________

        if(hat == 40){
            if(mask > 0) return 0;
            return 1;
        }
        if(dp[hat][mask] != -1) return dp[hat][mask];

        long long mod = 1e9+7;
        long long total = 0;
        for(int i=0;i<people;i++){
            if((mask&(1<<i)) == 0) continue;
            if(data[i][hat] == 1){
                long long int tempMask = mask;
                total += solve2(hats, dp, hat+1, tempMask ^ (1<<i), people, data);
                total %= mod;
            }
        }
        total += solve2(hats, dp, hat+1, mask, people, data);
        total %= mod;

        dp[hat][mask] = total%mod;
        return dp[hat][mask];
    }

    int numberWays(vector<vector<int>>& hats) {
        //_______________________________
        // FOR METHOD-1 

        // int people = hats.size();
        // long long int mask = (1<<10) - 1;
        // vector<vector<long long int>> dp(people+1, vector<long long int>((1<<10), -1));
        // return solve(hats, dp, 0, mask, people);
        //_______________________________
        int people = hats.size();
        long long int mask = (1<<people) -1 ;
        vector<vector<int>> data(10, vector<int>(41, 0));
        for(int i=0;i<people;i++){
            for(int j=0;j<hats[i].size();j++){
                data[i][hats[i][j]-1] = 1;
            }
        }
        vector<vector<long long int>> dp(41, vector<long long int>((1<<people), -1));
        return solve2(hats, dp, 0, mask, people, data);
    }
};
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