
"""simple docstring"""

def 							A						(    snake_case				:Union[str, Any]    , snake_case				:Optional[int]		)				->     Optional[Any]:
 __UpperCamelCase      =	[0 for i in range(r + 1		)]
 # nc0 = 1
 __UpperCamelCase      =	1
 for i in range(1    , n + 1		):
  # to compute current row from previous row.
  __UpperCamelCase      =	min(snake_case    , snake_case		)
  while j > 0:
   c[j] += c[j - 1]
   j -= 1
 return c[r]





print(binomial_coefficient(n=1_0, r=5))





"""simple docstring"""

def 							A						(    snake_case				:list[list[int]]    , snake_case				:int    , snake_case				:int    , snake_case				:list[int]		)				->     bool:

 # 1. Validate that path exists between current and next vertices
 if graph[path[curr_ind - 1]][next_ver] == 0:
  return False

 # 2. Validate that next vertex is not already in path
 return not any(vertex == next_ver for vertex in path		)






def 							A						(    snake_case				:list[list[int]]    , snake_case				:list[int]    , snake_case				:int		)				->     bool:

 # Base Case
 if curr_ind == len(snake_case		):
  # return whether path exists between current and starting vertices
  return graph[path[curr_ind - 1]][path[0]] == 1

 # Recursive Step
 for next_ver in range(0    , len(snake_case		)		):
  if valid_connection(snake_case    , snake_case    , snake_case    , snake_case		):
   # Insert current vertex  into path as next transition
   __UpperCamelCase      =	next_ver
   # Validate created path
   if util_hamilton_cycle(snake_case    , snake_case    , curr_ind + 1		):
    return True
   # Backtrack
   __UpperCamelCase      =	-1
 return False






def 							A						(    snake_case				:list[list[int]]    , snake_case				:int = 0		)				->     list[int]:
 __UpperCamelCase      =	[-1] * (len(snake_case		) + 1)
 # initialize start and end of path with starting index
 __UpperCamelCase      =	__UpperCamelCase      =	start_index
 # evaluate and if we find answer return path either return empty array
 return path if util_hamilton_cycle(snake_case    , snake_case    , 1		) else []



