AMC10 2018 B

AMC10 2018 B · Q5

AMC10 2018 B · Q5. It mainly tests Combinations, Inclusion–exclusion (basic).

How many subsets of $\lbrace2, 3, 4, 5, 6, 7, 8, 9\rbrace$ contain at least one prime number?

集合$\lbrace2, 3, 4, 5, 6, 7, 8, 9\rbrace$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）：满足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素数的子集数量；而不包含任何素数的子集数量等于集合 $\{4,6,8,9\}$ 的子集数量。含有 $n$ 个元素的集合有 $2^n$ 个子集，因此所求为 $2^8-2^4=256-16=240$。

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