<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:
   vector<vector<int>> dp;
int mod=1e9+7;
int func(int mask, int currhat, int n, vector<vector<int>> &nums, int len)
{
    if (currhat >= 40)
    {
        if (len == n)
            return 1;
        else
        {
            return -1;
        }
    }

    if (len == n)
        return 1;
    if (dp[currhat][mask] != -1)
        return dp[currhat][mask];
    int ans = 0;
    int p = func(mask, currhat + 1, n, nums, len);
    if (p != -1)
        ans = p;
    for (auto i : nums[currhat])
    {
        int num = (1 << i);
        if ((mask & num) == 0)
        {
            int x = func((mask | num), currhat + 1, n, nums, len + 1);
            if (x != -1)
            {
                ans =(ans+ x)%mod;
            }
        }
    }
    dp[currhat][mask]=ans;
    return ans;
}

int numberWays(vector<vector<int>> &hats)
{
    int n = hats.size();
    vector<vector<int>> nums(40);
    for (int i = 0; i < n; i++)
    {
        for (auto j : hats[i])
        {
            int hat = j - 1;
            nums[hat].push_back(i);
        }
    }

    dp.resize(40, vector<int>(1 << 10, -1));
    return func(0,0,n,nums,0);

}
};
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