2018 AMC 12B 第 5 题

Question

集合

\{2, 3, 4, 5, 6, 7, 8, 9\}

有多少个子集至少包含一个质数？

How many subsets of

\{2, 3, 4, 5, 6, 7, 8, 9\}

contain at least one prime number?

128

192

224

240

256

先数所有子集，再减去不含质数的子集。

Count all subsets, then subtract those containing no prime number

非质数为

\{4,6,8,9\}

，所以不含质数的子集就是这个集合的子集。

The non-primes are

\{4,6,8,9\},

so subsets with no prime number are subsets of that set

该集合有

8

个元素，共有

2^8=256

个子集。不含质数的子集只能使用四个非质数

\{4,6,8,9\}

，共有

2^4=16

个。

因此至少含一个质数的子集数为

256-16=240

。

所以正确答案是 D。

The set has

8

elements, giving

2^8=256

subsets. The subsets with no prime use only the four non-primes

\{4,6,8,9\},

and there are

2^4=16

of these.

So the number containing at least one prime is

256-16=240.

Thus, the correct answer is D.

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