AMC 8 2015 Problem 13

AMC8 2015 年第 13 题

基本计数原理★★★☆☆

How many subsets of two elements can be removed from the set

\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}

so that the mean (average) of the remaining numbers is 6?

有多少个两元素的子集可以从集合｛1,2,3,4,5,6,7,8,9,10,11｝中移走，使得集合中剩下的元素的平均值是 6？

A. $\text{ 1}$
B. $\text{ 2}$
C. $\text{ 3}$
D. $\text{ 5}$
E. $\text{ 6}$

正确答案 · Correct Answer

解析 · Solution

Solution 1

Since there will be 9 elements after removal, and their mean is 6, we know their sum is 54. We also know that the sum of the set pre-removal is 66. Thus, the sum of the 2 elements removed is 66-54=12. There are only subsets of 2 elements that sum to 12:

\{1,11\}, \{2,10\}, \{3, 9\}, \{4, 8\}, \{5, 7\}

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