<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 = pow(10, 9) + 7;

    void findSol(map<int, vector<int>> &mp, vector<vector<long long>> &dp, int currWear, int maxHat) {
        if(currWear == dp[0].size() - 1) {
            dp[maxHat][currWear] = 1;
            return;
        }

        auto it = mp.upper_bound(maxHat);
        if(it == mp.end()) {
            dp[maxHat][currWear] = 0;
            return;
        }
        long long posb = 0;
        int hat = (it -> first);
        for(int i = 0; i < (it -> second).size(); i++) {
            int nxtWearer = (it -> second)[i];
            if((currWear & (1 << nxtWearer)) == 0) {
                if(dp[hat][currWear | (1 << nxtWearer)] == -1) {
                    findSol(mp, dp, currWear | (1 << nxtWearer), hat);
                }
                posb = (posb + dp[hat][currWear | (1 << nxtWearer)]) % mod;
            }
        }
        if(dp[hat][currWear] == -1) {
            findSol(mp, dp, currWear, hat);
        }
        posb = (posb + dp[hat][currWear]) % mod;
        dp[maxHat][currWear] = posb;
    } 

    int numberWays(vector<vector<int>>& hats) {
        map<int, vector<int>> mp;
        vector<vector<long long>> dp(41, vector<long long>(pow(2, hats.size()), -1));
        for(int i = 0; i < hats.size(); i++) {
            for(int j = 0; j < hats[i].size(); j++) {
                mp[hats[i][j]].push_back(i);
            }
        }

        findSol(mp, dp, 0, 0);
        return dp[0][0];
        
    }
};
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