AMC12 2002 B

AMC12 2002 B · Q10

AMC12 2002 B · Q10. It mainly tests Basic counting (rules of product/sum), Remainders & modular arithmetic.

How many different integers can be expressed as the sum of three distinct members of the set $\{1,4,7,10,13,16,19\}$?

有多少个不同的整数可以表示为集合 $\{1,4,7,10,13,16,19\}$ 中三个不同成员之和？

(A) 13 13

(B) 16 16

(D) 30 30

(E) 35 35

Answer

Correct choice: (A)

正确答案：（A）

Solution

Subtracting 10 from each number in the set, and dividing the results by 3, we obtain the set $\{-3, -2, -1, 0, 1, 2, 3\}$. It is easy to see that we can get any integer between $-6$ and $6$ inclusive as the sum of three elements from this set, for the total of $\boxed{\mathrm{(A) } 13}$ integers.

将集合中每个数都减去 $10$，再把结果除以 $3$，得到集合 $\{-3, -2, -1, 0, 1, 2, 3\}$。容易看出，用该集合中三个元素之和可以得到从 $-6$ 到 $6$（含端点）的任意整数，共有 $\boxed{\mathrm{(A) } 13}$ 个整数。

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