2018 AMC 10B 第 5 题

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

数不含质数的子集，再从所有子集中减去。

Count the subsets with no prime and subtract from the total number of subsets

非质数元素是

4, 6, 8, 9

；只由它们组成的任意子集都不含质数。

The non-prime elements are

4, 6, 8, 9;

any subset of just those contains no prime

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.

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