Question

集合 $\{2, 3, 4, 5, 6, 7, 8, 9\}$ 的子集中，有多少个至少包含一个质数？

How many subsets of $\{2, 3, 4, 5, 6, 7, 8, 9\}$ contain at least one prime number?

$128$

$192$

$224$

$240$

$256$

答案：D

知识点：子集补集计数质数

难度评级：1200

解答：

反过来计数。这个集合共有 $2^8 = 256$ 个子集。不含任何质数的子集只能从非质数 $\{4, 6, 8, 9\}$ 中选出，共 $2^4 = 16$ 个。所以至少含一个质数的子集有 $256 - 16 = 240$ 个。正确答案是 D。

Count the complement. The set has $2^8 = 256$ subsets total. A subset avoids every prime exactly when it sticks to the non-primes $\{4, 6, 8, 9\},$ and there are $2^4 = 16$ of those. So $256 - 16 = 240$ subsets contain at least one prime. Thus, D is the correct answer.

2018 AMC 10B 第 5 题

答案：D

难度评级：1200

解答：

其他年份的第 5 题