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.

```python

USER_FILENAME = "<user_code>"
import sys, json, builtins
import math, heapq, bisect, itertools, functools, collections, statistics, fractions, decimal, operator, array, random, re, string, os, pathlib
try:
    import resource as _res
except Exception:
    _res = None
from collections import Counter, defaultdict, deque, OrderedDict
from functools import lru_cache, reduce
from itertools import accumulate, permutations, combinations, product
from math import gcd, ceil, floor, sqrt
from decimal import Decimal
try:
    from typing import *
except Exception:
    pass
try:
    sys.setrecursionlimit(1_000_000)
except Exception:
    pass
try:
    import site as _site
    try:
        _sys_sites = _site.getsitepackages() if hasattr(_site, "getsitepackages") else []
        for _p in _sys_sites:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
    try:
        _usp = _site.getusersitepackages()
        if isinstance(_usp, str):
            _usp = [_usp]
        for _p in _usp:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
except Exception:
    pass

def _disabled_input(*args, **kwargs):
    raise RuntimeError("input() is disabled in callable mode")
builtins.input = _disabled_input

ns = {"__name__": "__judge__"}  # prevent if __name__ == "__main__" blocks
for _k, _v in list(globals().items()):
    if not _k.startswith("_"):
        ns[_k] = _v
_CODE = 'class Solution:\n    # def numberWays(self, hats: List[List[int]]) -> int:\n        # const = 10**9+7\n        # cache = {}\n        # matr = [[False]*len(hats) for i in range(40)]\n        # maxx = -1\n        # used_h = [False]*40\n        # for j in range(len(hats)):\n        #     for i in hats[j]:\n        #         matr[i-1][j] = True\n        #         maxx = max(i-1, maxx)\n        #         used_h[i-1] = True\n        # def s(mask_p, h, n):\n        #     if n > h+1 or h < 0:\n        #         return 0\n        #     c = str(h)+mask_p\n        #     if c in cache:\n        #         return cache[c]\n        #     res = s(mask_p, h-1, n)\n        #     if used_h[h]:\n        #         for p in range(len(matr[h])):\n        #             if mask_p[p] == \\"1\\" and matr[h][p]:\n        #                 if n > 1:\n        #                     res += s(mask_p[:p] + \\"0\\" + mask_p[p+1:], h-1, n-1)\n        #                 else:\n        #                     res += 1\n        #     cache[c] = res%const\n        #     return cache[c]\n        # mask_p = \\"1\\"*len(hats)\n        # return s(mask_p, maxx, len(hats))%const\n\n\n        \n        \n        \n        \n        \n        \n        \n        \n        \n        \n        \n        \n        \n        \n        \n        \n        \n        \n        \n        \n        \n\n        \n        \n        \n        \n        \n        \n        \n        \n        \n        \n        \n\n    def _to_hat_to_people(self, person_to_hats):\n        min_hat, max_hat = None, None\n        for hats in person_to_hats:\n            for hat in hats:\n                if min_hat is None or hat < min_hat:\n                    min_hat = hat\n                if max_hat is None or hat > max_hat:\n                    max_hat = hat\n\n        number_of_hats = max_hat - min_hat + 1\n\n        hats_to_people = [[] for _ in range(number_of_hats)]\n        for person in range(len(person_to_hats)):\n            for hat in person_to_hats[person]:\n                hats_to_people[hat - min_hat].append(person)\n        return hats_to_people\n\n    def _number_of_ways(self, hat_to_people, person_to_hat_presence, hat, cache):\n        if all(person_to_hat_presence):\n            return 1\n        if hat >= len(hat_to_people):\n            return 0\n        key = hat, tuple(person_to_hat_presence)\n        if key in cache:\n            return cache[key]\n        else:\n            sum = 0\n            for person in hat_to_people[hat]:\n                if not person_to_hat_presence[person]:\n                    updated_person_to_hat_presence = person_to_hat_presence.copy()\n                    updated_person_to_hat_presence[person] = True\n                    sum += self._number_of_ways(hat_to_people, updated_person_to_hat_presence, hat + 1, cache)\n            sum += self._number_of_ways(hat_to_people, person_to_hat_presence, hat + 1, cache)\n            cache[key] = sum%(10**9+7)\n            return cache[key]\n\n    def numberWays(self, person_to_hats):\n        hat_to_people = self._to_hat_to_people(person_to_hats)\n        return self._number_of_ways(hat_to_people, [False] * len(person_to_hats), 0, {})'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'numberWays'
if _fn_name:
    if "Solution" in ns:
        try:
            _obj = ns["Solution"]()
            if hasattr(_obj, _fn_name):
                _target = getattr(_obj, _fn_name)
        except Exception:
            _target = None
    if _target is None and _fn_name in ns and callable(ns[_fn_name]):
        _target = ns[_fn_name]
# Fallback: if there's exactly one user-defined function, use it
if _target is None:
    _cands = [v for k, v in ns.items() if callable(v) and getattr(v, "__module__", "") in ("__judge__", "__main__")]
    _user_funcs = [c for c in _cands if getattr(c, "__name__", "").startswith("_") is False]
    if len(_user_funcs) == 1:
        _target = _user_funcs[0]
if _target is None:
    raise RuntimeError(f"Could not resolve callable '{_fn_name}'")

_raw = sys.stdin.read().strip()
if _raw:
    _parsed = json.loads(_raw)
    if isinstance(_parsed, dict):
        _kwargs = _parsed.get("kwargs", {})
        _args = _parsed.get("args", [])
    else:
        _args, _kwargs = _parsed, {}
else:
    _args, _kwargs = [], {}
if not isinstance(_args, (list, tuple)):
    _args = [_args]
_res = _target(*_args, **_kwargs)
try:
    sys.stdout.write(repr(_res) + "\n")
except Exception:
    pass

```

Measured memory_bytes: 8550