AMC12 2018 B

AMC12 2018 B · Q5

AMC12 2018 B · Q5. It mainly tests Basic counting (rules of product/sum), Primes & prime factorization.

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

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

(A) 128 128

(B) 192 192

(D) 240 240

(E) 256 256

Answer

Correct choice: (D)

正确答案：（D）

Solution

Answer (D): The number of qualifying subsets equals the difference between the total number of subsets of $\{2, 3, 4, 5, 6, 7, 8, 9\}$ and the number of such subsets containing no prime numbers, which is the number of subsets of $\{4, 6, 8, 9\}$. A set with $n$ elements has $2^n$ subsets, so the requested number is $2^8 - 2^4 = 256 - 16 = 240$.

答案（D）：符合条件的子集数量等于集合 $\{2, 3, 4, 5, 6, 7, 8, 9\}$ 的子集总数减去不包含任何质数的子集数量；而不包含质数的子集正是集合 $\{4, 6, 8, 9\}$ 的子集数。含有 $n$ 个元素的集合有 $2^n$ 个子集，因此所求数量为 $2^8 - 2^4 = 256 - 16 = 240$。

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