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 = set()\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.add(i-1)\n#         def s(mask_p, h, n):\n#             if h < 0 or n > h+1:\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 h in used_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 res%const\n#         mask_p = \\"1\\"*len(hats)\n#         return s(mask_p, maxx, len(hats))%const\n    \n    \nclass Solution:\n    def numberWays(self, hats: List[List[int]]) -> int:\n        const = 10**9+7\n        matr = [0 for i in range(40)]\n        maxx = -1\n        used_h = set()\n        for j in range(len(hats)):\n            for i in hats[j]:\n                matr[i-1] += 2**(j)\n                maxx = max(i-1, maxx)\n                used_h.add(i-1)\n        matr = matr[:maxx+1]\n        mask_p = 2**len(hats)-1\n        def s(mask_p, n, h, c_h, cache):\n            if n > c_h:\n                return 0\n            if cache[mask_p][h] != None:\n                return cache[mask_p][h]\n            if mask_p == 0:\n                return 1\n            res = 0\n            if h in used_h:\n                c_h -= 1\n                b = mask_p & matr[h]\n                i = 0\n                while b > 0:\n                    if b % 2 == 0:\n                        b /= 2\n                    else:\n                        b -= 1\n                        b /= 2\n                        res += s(mask_p - 2**(i), n-1, h-1, c_h, cache)\n                    i += 1\n                # b = bin(mask_p & matr[h])[2:]\n                # for p in range(len(b)):\n                #     if b[p] == \\"1\\":\n                #         res += s(mask_p - 2**(len(b)-p-1), n-1, h-1, c_h, cache)\n            res += s(mask_p, n, h-1, c_h, cache)\n            cache[mask_p][h] = res%const\n            return cache[mask_p][h]\n        return s(mask_p, len(hats), maxx, len(used_h), [[None]*(maxx+1) for i in range(mask_p+1)])'
_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: 10180