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question:def smallest_rectangle(L: int, M: int, positions: List[Tuple[int, int]]) -> Tuple[int, int]: Given the side length L of the garden and M positions of rare flowering plants, returns the dimensions of the smallest rectangle that can enclose all the rare plants after any number of row transitions. Parameters: L (int): the side length of the garden. M (int): the number of rare flowering plants. positions (list of tuples): the positions of the rare flowering plants. Returns: tuple: the dimensions of the smallest rectangle (height, width). >>> smallest_rectangle(5, 3, [(1, 2), (3, 3), (4, 1)]) (4, 3) >>> smallest_rectangle(6, 4, [(2, 2), (2, 4), (5, 1), (6, 5)]) (5, 5) >>> smallest_rectangle(4, 1, [(1, 1)]) (1, 1) >>> smallest_rectangle(5, 3, [(1, 2), (3, 2), (5, 2)]) (5, 1) >>> smallest_rectangle(5, 3, [(1, 1), (1, 2), (1, 5)]) (1, 5) >>> smallest_rectangle(5, 3, [(2, 2), (2, 3), (2, 4)]) (1, 3) # Initialize the min and max bounds for rows and columns # Your solution goes here

answer:def smallest_rectangle(L, M, positions): Given the side length L of the garden and M positions of rare flowering plants, returns the dimensions of the smallest rectangle that can enclose all the rare plants after any number of row transitions. Parameters: L (int): the side length of the garden. M (int): the number of rare flowering plants. positions (list of tuples): the positions of the rare flowering plants. Returns: tuple: the dimensions of the smallest rectangle (height, width). # Initialize the min and max bounds for rows and columns min_x = L max_x = 0 min_y = L max_y = 0 # Traverse through all positions to find the min and max bounds for x, y in positions: if x < min_x: min_x = x if x > max_x: max_x = x if y < min_y: min_y = y if y > max_y: max_y = y # Rectangle dimensions height = max_x - min_x + 1 width = max_y - min_y + 1 return height, width

question:from typing import List def is_pack_cheaper(item_prices: List[int], pack_price: int) -> bool: Determines if buying a pack of 5 items at a discounted price is cheaper than buying the five cheapest items individually. Parameters: item_prices (List[int]): A list of individual item prices. pack_price (int): The discounted price for a pack of 5 items. Returns: bool: True if the pack price is cheaper, False otherwise. >>> is_pack_cheaper([2, 3, 1, 5, 4, 6, 7, 8], 10) True >>> is_pack_cheaper([2, 3, 1, 5, 4, 6, 7, 8], 15) False >>> is_pack_cheaper([2, 3, 1, 5, 4, 6, 7, 8], 16) False >>> is_pack_cheaper([1, 2, 3, 4, 5], 14) True >>> is_pack_cheaper([100, 200, 3, 4, 5, 2, 1, 300], 10) True

answer:from typing import List def is_pack_cheaper(item_prices: List[int], pack_price: int) -> bool: Determines if buying a pack of 5 items at a discounted price is cheaper than buying the five cheapest items individually. Parameters: item_prices (List[int]): A list of individual item prices. pack_price (int): The discounted price for a pack of 5 items. Returns: bool: True if the pack price is cheaper, False otherwise. # Sort the list of item prices to find the five cheapest items item_prices.sort() # Sum the five cheapest items sum_of_cheapest_five = sum(item_prices[:5]) # Compare the sum with the pack price return pack_price < sum_of_cheapest_five

question:def can_split_into_patterns(t: int, cases: list) -> list: Given a list of integers, check if it is possible to split the list into exactly k non-empty continuous subsequences such that each subsequence is a pattern of consecutive numbers that increase by 1. -----Input----- The input consists of multiple test cases. The first line contains a single integer t (1 le t le 50) — the number of test cases. The first line of each test case description contains two integers n and k (1 le n le 100, 1 le k le n) — the length of the list and the number of patterns required. The second line of each test case description contains n space-separated integers a_1, a_2, ldots, a_n (1 le a_i le 1000) — the list of integers. -----Output----- For each test case, return "YES" (without quotes) if it is possible to split the list into exactly k patterns, and "NO" (without quotes) otherwise. >>> can_split_into_patterns(4, [ ... ((6, 2), [1, 2, 3, 5, 6, 7]), ... ((5, 1), [3, 4, 5, 6, 7]), ... ((7, 3), [10, 11, 12, 1, 2, 3, 8]), ... ((4, 4), [4, 5, 6, 7]) ... ]) ["YES", "YES", "YES", "NO"] >>> can_split_into_patterns(3, [ ... ((1, 1), [1]), ... ((2, 2), [1, 3]), ... ((3, 1), [5, 6, 7]) ... ]) ["YES", "YES", "YES"]

answer:def can_split_into_patterns(t, cases): results = [] for i in range(t): n, k = cases[i][0] nums = cases[i][1] # Step 1: Find the lengths of all continuous increasing subsequences lengths = [] current_length = 1 for j in range(1, n): if nums[j] == nums[j - 1] + 1: current_length += 1 else: lengths.append(current_length) current_length = 1 lengths.append(current_length) # Step 2: Check if we can use the found lengths to get exactly k patterns if len(lengths) < k: results.append("NO") else: if len(lengths) == k: results.append("YES") else: remaining_splits_needed = k - len(lengths) # Check if we have enough larger subsequences to be split further splittable_count = sum(l - 1 for l in lengths if l > 1) if splittable_count >= remaining_splits_needed: results.append("YES") else: results.append("NO") return results

question:from itertools import permutations from typing import List, Tuple def generate_permutations(A: List[str], r: int) -> List[Tuple[str, ...]]: Generates all r-length permutations of the list A. Parameters: A (list): List of characters. r (int): Length of the permutations. Returns: list: List of r-length permutations. >>> generate_permutations(['A', 'B', 'C'], 2) [('A', 'B'), ('A', 'C'), ('B', 'A'), ('B', 'C'), ('C', 'A'), ('C', 'B')] >>> generate_permutations(['1', '2', '3'], 3) [('1', '2', '3'), ('1', '3', '2'), ('2', '1', '3'), ('2', '3', '1'), ('3', '1', '2'), ('3', '2', '1')] >>> generate_permutations(['a', 'b'], 2) [('a', 'b'), ('b', 'a')] >>> generate_permutations(['x'], 1) [('x',)] >>> generate_permutations(['A', 'B', 'C', 'D'], 3) [(('A', 'B', 'C'), ('A', 'B', 'D'), ('A', 'C', 'B'), ('A', 'C', 'D'), ('A', 'D', 'B'), ('A', 'D', 'C'), ('B', 'A', 'C'), ('B', 'A', 'D'), ('B', 'C', 'A'), ('B', 'C', 'D'), ('B', 'D', 'A'), ('B', 'D', 'C'), ('C', 'A', 'B'), ('C', 'A', 'D'), ('C', 'B', 'A'), ('C', 'B', 'D'), ('C', 'D', 'A'), ('C', 'D', 'B'), ('D', 'A', 'B'), ('D', 'A', 'C'), ('D', 'B', 'A'), ('D', 'B', 'C'), ('D', 'C', 'A'), ('D', 'C', 'B')] ]

answer:from itertools import permutations def generate_permutations(A, r): Generates all r-length permutations of the list A. Parameters: A (list): List of characters. r (int): Length of the permutations. Returns: list: List of r-length permutations. return list(permutations(A, r))