<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, person_to_hats: List[List[int]], mod=10 ** 9 + 7) -> int:
    n = len(person_to_hats)
    m = -1
    for hats in person_to_hats:
      if not hats:
        return 0
      m = max(m, max(hats))

    # Use m+1 because m is 1 based in the data `person_to_hats`.
    dp = [[-1 for _ in range(m + 1)] for _ in range(1 << n)]
    hat_to_persons = [[] for _ in range(m + 1)]

    # hat_to_persons[hat_number] contains all persons that can wear this hat.
    for person, hat_list in enumerate(person_to_hats):
      for hat in hat_list:
        hat_to_persons[hat].append(person)

    done_mask = (1 << n) - 1

    def helper(mask, hat):
      if mask == done_mask:
        return 1
      if hat > m:
        return 0

      if dp[mask][hat] != -1:
        return dp[mask][hat]

      # Number of ways when the given hat is not chosen by any person
      cnt = helper(mask, hat + 1)
      cnt %= mod

      # Iterate over all the persons who can choose the given hat
      for person in hat_to_persons[hat]:
        # Continue if the person already has a hat
        if mask & (1 << person):
          continue
        # Otherwise, set the bit for the person, indicating they've been assigned a hat
        cnt += helper(mask | (1 << person), hat + 1)
        cnt %= mod

      dp[mask][hat] = cnt
      return cnt

    return helper(0, 1)
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