<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 numberWays(vector<vector<int>>& hats) {
        int n = hats.size();
        int mask = (1 << n)- 1; // 1024

        vector<vector<int>> hat_people(40);
        for (int i = 0; i < hats.size(); ++i) {
            for (int hat : hats[i]) {
                hat_people[hat-1].push_back(i);
            }
        }

        vector<vector<int>> dp(mask+1, vector<int>(40,-1)); // visited_people -> hat

        return dfs(hat_people, dp, mask, 0);
    }

    int dfs(vector<vector<int>>& hat_people, vector<vector<int>>& dp, int people_mask, int hat) {
        if (people_mask == 0) return 1; // every one is wearing hat
        if (hat >= 40) return 0;
        if (dp[people_mask][hat] != -1) return dp[people_mask][hat];

        // dont wear this hat
        int dont_choose = dfs(hat_people, dp, people_mask, hat+1);

        // wear
        int choose = 0;
        for (int person : hat_people[hat]) {
            if (people_mask & (1 << person)) {
                int newMask = people_mask;
                newMask = newMask & ~(1 << person);
                int check = dfs(hat_people, dp, newMask, hat+1);
                choose = (choose + check)%MOD;
            }
        }

        return dp[people_mask][hat] = (choose + dont_choose)%MOD;
    }
};
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