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 = 'from functools import lru_cache\n\np = 10**9 +7 \nclass Solution:\n    \n#     def get_number_of_ways(self, hats, mask, p_id): #p_id = person_id\n#         if p_id>=len(hats):\n#             return 1\n        \n#         if dp[p_id].get(mask, None):\n#             return dp[p_id][mask]\n#         else:\n#             dp[p_id][mask] = 0\n        \n#         for i in hats[p_id]:\n#             if (mask & (1<<i))>>i :\n#                 continue\n#             temp_mask = mask | (1<<i)\n#             val = self.get_number_of_ways(hats, temp_mask, p_id+1)\n#             if val == -1:\n#                 continue\n            \n#             dp[p_id][mask] += val\n        \n#         return dp[p_id][mask]\n\n        \n    def numberWays(self, hats):\n        max_hat_id = 0\n        distinct_hats = set()\n        for person_hats in hats:\n            for i in range(len(person_hats)):\n                person_hats[i] -= 1\n                distinct_hats.add( person_hats[i])\n        hat_id_mapping = dict()\n        i = 0\n        for hat in distinct_hats:\n            hat_id_mapping[hat] = i\n            i += 1\n            \n        persons = [[] for i in range(len(distinct_hats))]\n        \n        for p_id in range(len(hats)):\n            for hat_id in hats[p_id]:\n                persons[hat_id_mapping[hat_id]].append(p_id)\n        dp = dict()\n        for i in range(len(persons)):\n            dp[i]= dict()\n        allmask = (1<<len(hats))-1\n        \n        def get_number_of_ways( mask, hat_id): #p_id = person_id\n            if mask == allmask :\n                return 1\n            if hat_id>=len(persons):\n                return 0\n            \n            if (hat_id, mask) in dp:\n                return dp[(hat_id, mask)]\n            else:\n                dp[(hat_id, mask)] = get_number_of_ways( mask, hat_id+1)\n            for i in persons[hat_id]:\n                if (mask & (1<<i))>>i :\n                    continue\n                temp_mask = mask | (1<<i)\n                val = get_number_of_ways( temp_mask, hat_id+1)\n                \n                dp[(hat_id, mask)] += val\n            \n            return dp[(hat_id, mask)]\n            \n        \n        ans = get_number_of_ways( 0, 0)\n        return ans%p\n        \n        \n        \n        \n        \n        # @lru_cache(None)\n#         def get_number_of_ways( mask, hat_id, allmask): #p_id = person_id\n#             if mask == allmask :\n#                 return 1\n#             if hat_id>=len(persons):\n#                 return 0\n            \n#             if dp[hat_id].get(mask, None):\n#                 return dp[hat_id][mask]\n#             else:\n#                 dp[hat_id][mask] = get_number_of_ways( mask, hat_id+1, allmask)\n#             for i in persons[hat_id]:\n#                 if (mask & (1<<i))>>i :\n#                     continue\n#                 temp_mask = mask | (1<<i)\n#                 val = get_number_of_ways( temp_mask, hat_id+1, allmask)\n                \n#                 dp[hat_id][mask] += val\n            \n#             return dp[hat_id][mask]\n            \n        \n        \n#         return get_number_of_ways( 0, 0, allmask) % p\n'
_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: 8970