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There are `n` people and 40 types of hats labeled from 1 to 40.

Given a list of list of integers `hats`, where `hats[i]` is a list of all hats preferred by the `i-th` person.

Return the number of ways that the n people wear different hats to each other.

Since the answer may be too large, return it modulo `10^9 + 7`.

Example 1:

```Input: hats = [[3,4],[4,5],[5]]
Output: 1
Explanation: 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.```

Example 2:

```Input: hats = [[3,5,1],[3,5]]
Output: 4
Explanation: There are 4 ways to choose hats
(3,5), (5,3), (1,3) and (1,5)
```

Example 3:

```Input: hats = [[1,2,3,4],[1,2,3,4],[1,2,3,4],[1,2,3,4]]
Output: 24
Explanation: Each person can choose hats labeled from 1 to 4.
Number of Permutations of (1,2,3,4) = 24.
```

Example 4:

```Input: hats = [[1,2,3],[2,3,5,6],[1,3,7,9],[1,8,9],[2,5,7]]
Output: 111
```

Constraints:

• `n == hats.length`
• `1 <= n <= 10`
• `1 <= hats[i].length <= 40`
• `1 <= hats[i][j] <= 40`
• `hats[i]` contains a list of unique integers.

class Solution { public int numberWays(List<List<Integer>> hats) { } }