Question

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?

$\text{ 1}$

$\text{ 2}$

$\text{ 3}$

$\text{ 5}$

$\text{ 6}$

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Written solution:

The original set has sum $1+2+\cdots+11=66$ . After removing two numbers, $9$ numbers remain and must have mean $6$ , so their sum must be $9\cdot6=54$ .

Therefore the two removed numbers must have sum $66-54=12$ . The possible two-element subsets are $\{1,11\},\{2,10\},\{3,9\},\{4,8\},\{5,7\}$ , so there are $5$ choices.

Thus, D is the correct answer.

2015 AMC 8 Problem 13

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Written solution:

Problem 13 in Other Years