My first idea is depth-first-search: iterate all people, try to give them different hats. The solution got TLE (Time Limit Exceeded). Then as a hint from discussion forum, I started to iterate hat (instead of people), try to give them different people. The solution also got TLE (even I used lru_cache for function):

```from collections import defaultdict

class Solution:

def numberWays(self, hats: List[List[int]]) -> int:
hp = defaultdict(set)
for index, hat in enumerate(hats):
for _id in hat:

hp = [people for people in hp.values()]
@functools.lru_cache(None)
def dfs(start, path) -> int:
if len(path) == len(hats):
return 1
if start == len(hp):
return 0
total = 0
for person in (hp[start] - set(path)):
total += dfs(start + 1, tuple(list(path) + [person]))
total += dfs(start + 1, path)

return dfs(0, tuple())```

Using list as data structure to record visited node is not efficient enough in this case. Since there will be no more than 10 people, the most efficient data structure to record visited people is bits.

My final solution is still using dfs (by using lru_cache, it is also a dynamic-programming):

```from collections import defaultdict

class Solution:

def numberWays(self, hats: List[List[int]]) -> int:
hp = defaultdict(set)
for index, hat in enumerate(hats):
for _id in hat:

hp = [people for people in hp.values()]
@functools.lru_cache(None)
def dfs(start, mask) -> int:
if bin(mask).count('1') == len(hats):
return 1
if start == len(hp):
return 0
total = 0
for person in hp[start]:
if (1 << person) & mask > 0:
continue
mask |= 1 << person
total += dfs(start + 1, mask)
mask ^= 1 << person
total += dfs(start + 1, mask)