<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 count_ways(int cap,int left_persons,vector<int> per_hav_hat[],vector<int> &avl_caps,vector<vector<int>> &dp)
    {
        int N = 1000000007;
        if(left_persons==0)
        {
            return 1;
        }
        if(cap>40)
        {
            return 0;
        }

        if(dp[cap][left_persons]!=-1)
        {
            return dp[cap][left_persons];
        }

        int count = 0;
        if(avl_caps[cap]!=0)
        {
            for(auto person:per_hav_hat[cap])
            {
                int avl =  left_persons & (1<<person);
                if(avl)
                {
                    int person_left = left_persons ^ (1<<person);
                    count = (count + count_ways(cap+1,person_left,per_hav_hat,avl_caps,dp))%N;
                }
            }
        }
        count = (count + count_ways(cap + 1,left_persons,per_hav_hat,avl_caps,dp))%N;

        return dp[cap][left_persons] = count%N;
    }
    int numberWays(vector<vector<int>>& hats) {
        int n = hats.size();
        vector<int> per_hav_hat[41];
        vector<int> cap_exist(41,0);
        for(int i=0;i<n;i++)
        {
            int sz = hats[i].size();
            for(int j=0;j<sz;j++)
            {
                int cap = hats[i][j];
                per_hav_hat[cap].push_back(i);
                cap_exist[cap]=1;
            }
        }
        int tot_per = pow(2,n)-1;
        vector<vector<int>> dp(41,vector<int> (tot_per+1,-1));
        return count_ways(1,tot_per,per_hav_hat,cap_exist,dp);
    }
};
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