<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 ll long long int
#define vi vector<int>
#define vvi vector<vi>
#define vll vector<ll>
#define vvll vector<vll>
#define mod 1000000007
#define total 40

/*
    dp[i][mask] represents the number of ways to wear different hats
        when upto ith hat has been processed for persons marked in the mask

    mask is over people who have been assigned a hat
*/
class Solution {
    ll recur(vvi &arr, ll mask, int i, vvll &dp, int &n){
        if(mask == (1<<n)-1)return 1;

        if(i == -1)return 0;

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

        ll ans = recur(arr, mask, i-1, dp, n);
        for(int j:arr[i]){
            if(!(mask&(1<<j))){
                ans = (ans + recur(arr, mask|(1<<j), i-1, dp, n))%mod;
            }
        }
        return dp[i][mask] = ans;
    }
public:
    int numberWays(vvi& arr) {
        int n = (arr).size();

        vvll dp(total, vll(1<<n, -1));
        vvi dp1(total);
        for(int i=0;i<n;i++){
            for(int j:arr[i]){
                dp1[j-1].push_back(i);
            }
        }
        return recur(dp1, 0, total-1, dp, n);
    }
};
15000