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#         htop = [[] for i in range(41)] # htop: hat to people list\n#         for p, prefer_hats in enumerate(hats):\n#             for h in prefer_hats:\n#                 htop[h].append(p)\n#         htop = list(filter(lambda h: h, htop)) # filter out hats no one wants\n        \n#         num_hats, num_people = len(htop), len(hats)\n#         if num_hats < num_people:\n#             return 0\n#         print(htop)\n#         MOD = 10**9+7\n#         @functools.lru_cache(None)\n#         def dp(i, mask):\n#             if bin(mask).count('1') == num_people:\n#                 return 1\n#             if i == num_hats:\n#                 return 0\n#             res = dp(i+1, mask) # not using the current hat\n#             for p in htop[i]:\n#                 if mask & (1<<p) == 0:\n#                     mask |= 1<<p\n#                     res += dp(i+1, mask)\n#                     mask ^= 1<<p\n#             return res%MOD\n#         return dp(0, 0)\nclass Solution:\n    def numberWays(self, hats: List[List[int]]) -> int:        \n        hat2ppl = defaultdict(set)        \n        for p, hats_i in enumerate(hats):\n            for h in hats_i: hat2ppl[h].add(p)\n        if len(hats) > len(hat2ppl): return 0 \n        M = 10**9 + 7\n        @lru_cache(None)\n        def dfs(h, mask):\n            if h == 42: return 0            \n            if bin(mask).count('1') == len(hats): return 1            \n            ans = dfs(h+1, mask)\n            for p in hat2ppl[h]:\n                x = 1 << p\n                if mask & x == 0:\n                    mask |= x\n                    ans += dfs(h+1, mask)\n                    mask ^= x\n            return ans % M\n        return dfs(1, 0) % M"
_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: 6264