2025 AMC 12A 第 13 题

Question

令

C = \{1, 2, 3, \ldots, 13\}

。令

N

为最大的整数，使得存在一个有

N

个元素的

C

的子集，并且该子集不含五个连续整数。现在从

C

中随机不放回地选出

N

个整数。所选元素不包含五个连续整数的概率是多少？

Let

C = \{1, 2, 3, \ldots, 13\}.

Let

N

be the greatest integer such that there exists a subset of

C

with

N

elements that does not contain five consecutive integers. Suppose

N

integers are chosen at random from

C

without replacement. What is the probability that the chosen elements do not include five consecutive integers?

\dfrac{3}{130}

\dfrac{3}{143}

\dfrac{5}{143}

\dfrac{1}{26}

\dfrac{5}{78}

答案：D

知识点：组合基本概率极端原理

难度评级：1660

小提示：

去掉两个合适的元素就能打断每一段五连整数，所以

N = 11

。

Removing two well-placed elements breaks every run of five, so

N = 11

大提示：

选

11

个等价于去掉

2

个；数一数哪两个被去掉的元素能碰到每一段五个连续整数

Choosing

11

means removing

2

; count removals whose two elements hit every block of five consecutive integers

解答：

要避免五个连续整数，只需去掉两个元素（例如

5

和

10

），而去掉一个元素不可能同时击中两个不相交的区块

\{1,\ldots,5\}

和

\{9,\ldots,13\}

。因此

N = 11

。

从

13

个元素中选

11

个，等价于去掉

2

个，共有

\binom{13}{2} = 78

种。所选集合不含五个连续整数，当且仅当两个被去掉的元素合起来与每个窗口

\{t, t+1, t+2, t+3, t+4\}

相交，其中

t = 1, \ldots, 9

。

这迫使一个被去掉的元素在

\{1,\ldots,5\}

中，另一个在

\{9,\ldots,13\}

中，并且两者相距不超过

5

。有效的去法为

\{4,9\}

、

\{5,9\}

和

\{5,10\}

，共

3

种。

概率为

\dfrac{3}{78} = \dfrac{1}{26}

。

因此，正确答案是 D。

To avoid five consecutive integers, it suffices to remove two elements (for example

5

and

10

), and no single removal can hit both disjoint blocks

\{1,\ldots,5\}

and

\{9,\ldots,13\}.

Thus

N = 11.

Choosing

11

of

13

elements is the same as removing

2,

which can be done in

\binom{13}{2} = 78

ways. The chosen set avoids five consecutive integers exactly when the two removed elements together intersect every window

\{t, t+1, t+2, t+3, t+4\}

for

t = 1, \ldots, 9.

This forces one removed element in

\{1,\ldots,5\},

the other in

\{9,\ldots,13\},

and the two within

5

of each other. The valid removals are

\{4,9\},

\{5,9\},

and

\{5,10\},

giving

3

of them.

The probability is

\dfrac{3}{78} = \dfrac{1}{26}.

Thus, the correct answer is D.

2025 AMC 12A 第 13 题

答案：D

难度评级：1660

小提示：

大提示：

解答：

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