<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:
    def numberWays(self, hats: List[List[int]]) -> int:
        n = len(hats)
        mod = 10 ** 9 + 7
        hat_nums = set()
        for h in hats: hat_nums |= set(h)
        hat_nums = sorted(hat_nums)
        n_hats = len(hat_nums)
        if n_hats < n: return 0
        hats_dict = {x: i for i, x in enumerate(hat_nums)}
        hat_people = [[] for _ in hat_nums]
        for i, lst in enumerate(hats):
            for j in lst:
                hat_people[hats_dict[j]].append(i)

        row = {0: 1}
        for i in range(n_hats):
            prev_row = row
            #row = dict(prev_row)
            row = {}
            for bm, f in prev_row.items():
                # Only include the bitmasks for which there
                # remain enough hats for every person to
                # get a hat
                if bm.bit_count() + n_hats - i > n:
                    row[bm] = row.get(bm, 0) + f
                for j in hat_people[i]:
                    bm2 = bm | (1 << j)
                    if bm2 == bm: continue
                    row[bm2] = (row.get(bm2, 0) + f) % mod
        return sum(row.values()) % mod
16700