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203 lines
5.7 KiB
Python
Executable file
203 lines
5.7 KiB
Python
Executable file
"""#
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### 谜题描述
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Andrey thinks he is truly a successful developer, but in reality he didn't know about the binary search algorithm until recently. After reading some literature Andrey understood that this algorithm allows to quickly find a certain number x in an array. For an array a indexed from zero, and an integer x the pseudocode of the algorithm is as follows:
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<image>
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Note that the elements of the array are indexed from zero, and the division is done in integers (rounding down).
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Andrey read that the algorithm only works if the array is sorted. However, he found this statement untrue, because there certainly exist unsorted arrays for which the algorithm find x!
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Andrey wants to write a letter to the book authors, but before doing that he must consider the permutations of size n such that the algorithm finds x in them. A permutation of size n is an array consisting of n distinct integers between 1 and n in arbitrary order.
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Help Andrey and find the number of permutations of size n which contain x at position pos and for which the given implementation of the binary search algorithm finds x (returns true). As the result may be extremely large, print the remainder of its division by 10^9+7.
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Input
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The only line of input contains integers n, x and pos (1 ≤ x ≤ n ≤ 1000, 0 ≤ pos ≤ n - 1) — the required length of the permutation, the number to search, and the required position of that number, respectively.
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Output
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Print a single number — the remainder of the division of the number of valid permutations by 10^9+7.
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Examples
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Input
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4 1 2
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Output
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6
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Input
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123 42 24
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Output
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824071958
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Note
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All possible permutations in the first test case: (2, 3, 1, 4), (2, 4, 1, 3), (3, 2, 1, 4), (3, 4, 1, 2), (4, 2, 1, 3), (4, 3, 1, 2).
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Here is a reference code to solve this task. You can use this to help you genereate cases or validate the solution.
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```python
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'''input
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123 42 24
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'''
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from bisect import bisect_right as bl
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from random import randint as R
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RI = lambda : [int(_x) for _x in raw_input().split()]
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mod = 10**9+7
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F = [1]*(1010)
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for i in range(1009):
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F[i+1]=(F[i]*(i+1))%mod
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n,x,p = RI()
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l = 0
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r = n
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big = n-x
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small = x-1
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ans = 1
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while l < r:
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mid = (l+r)/2
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if mid == p:
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l = mid + 1
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continue
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elif mid < p:
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ans = (ans*small)%mod
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small -= 1
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l = mid + 1
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else:
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ans = (ans*big)%mod
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big -= 1
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r = mid
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ans = (ans*F[(big+small)])%mod
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print ans
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```
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请完成上述谜题的训练场环境类实现，包括所有必要的方法。
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"""
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from bootcamp import Basebootcamp
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import random
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import re
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from bootcamp import Basebootcamp
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MOD = 10**9 + 7
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class Cbinarysearchbootcamp(Basebootcamp):
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def __init__(self, **params):
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super().__init__()
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self.mod = MOD
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# 预先计算阶乘数组，最多到2000
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self.max_fact = 2000
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self.fact = [1] * (self.max_fact + 1)
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for i in range(1, self.max_fact + 1):
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self.fact[i] = (self.fact[i-1] * i) % self.mod
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def compute_answer(self, n, x, pos):
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l = 0
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r = n
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big = n - x
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small = x - 1
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ans = 1
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while l < r:
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mid = (l + r) // 2
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if mid == pos:
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l = mid + 1
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continue
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elif mid < pos:
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ans = (ans * small) % self.mod
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small -= 1
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l = mid + 1
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else:
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ans = (ans * big) % self.mod
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big -= 1
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r = mid
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remaining = big + small
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if remaining >= 0:
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ans = (ans * self.fact[remaining]) % self.mod
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else:
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ans = 0
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return ans
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def case_generator(self):
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# Ensure x is between 1 and n, and pos is between 0 and n-1
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n = random.randint(1, 1000)
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x = random.randint(1, n)
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pos = random.randint(0, n - 1)
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correct_answer = self.compute_answer(n, x, pos)
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return {
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'n': n,
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'x': x,
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'pos': pos,
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'correct_answer': correct_answer
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}
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@staticmethod
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def prompt_func(question_case):
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n = question_case['n']
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x = question_case['x']
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pos = question_case['pos']
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prompt = f"给定n={n}, x={x}, pos={pos}。请计算满足以下条件的排列数目，结果模10^9+7。\n"
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prompt += "条件如下：\n"
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prompt += "1. 排列包含1到n的所有整数，每个整数恰好出现一次。\n"
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prompt += "2. 数字x位于位置pos（数组索引从0开始）。\n"
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prompt += "3. 使用以下二分查找实现能够找到x并返回True。\n"
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prompt += "二分查找的伪代码如下：\n"
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prompt += "函数binary_search(a, x):\n"
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prompt += " l = 0\n"
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prompt += " r = n\n"
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prompt += " while l < r:\n"
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prompt += " mid = (l + r) // 2\n"
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prompt += " if a[mid] <= x:\n"
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prompt += " l = mid + 1\n"
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prompt += " else:\n"
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prompt += " r = mid\n"
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prompt += " return a[l-1] == x\n"
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prompt += "请将答案放在[answer]和[/answer]标签之间，例如：\n"
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prompt += "答案：[answer]数值[/answer]"
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return prompt
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@staticmethod
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def extract_output(output):
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pattern = r'\[answer\](\d+)\[/answer\]'
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matches = re.findall(pattern, output)
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if matches:
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return matches[-1] # 返回最后一个匹配的结果
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else:
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return None
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@classmethod
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def _verify_correction(cls, solution, identity):
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if solution is None:
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return False
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expected = identity['correct_answer']
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try:
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solution_int = int(solution)
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return solution_int == expected
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except (ValueError, TypeError):
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return False
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