Difference between revisions of "2015 AMC 8 Problems/Problem 13"
(Created page with "How many subsets of two elements can be removed from the set <math>\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}</math> so that the mean (average) of the remaining numbers is 6? <mat...") |
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<math>\textbf{(A)}\text{ 1}\qquad\textbf{(B)}\text{ 2}\qquad\textbf{(C)}\text{ 3}\qquad\textbf{(D)}\text{ 5}\qquad\textbf{(E)}\text{ 6}</math> | <math>\textbf{(A)}\text{ 1}\qquad\textbf{(B)}\text{ 2}\qquad\textbf{(C)}\text{ 3}\qquad\textbf{(D)}\text{ 5}\qquad\textbf{(E)}\text{ 6}</math> | ||
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+ | ==Solution== | ||
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+ | Since there will be <math>9</math> elements after removal, and their mean is <math>6</math>, we know their sum is <math>54</math>. We also know that the sum of the set pre-removal is <math>66</math>. Thus, the sum of the <math>2</math> elements removed is <math>66-54=12</math>. There are only <math>5</math> subsets of <math>2</math> elements that sum to <math>12</math>: <math>\{1,11\}, \{2,10\}, \{3, 9\}, \{4, 8\}, \{5, 7\}</math>. Therefore, our answer is <math>\textbf{(D)}\text{ 5}</math>. |
Revision as of 16:12, 25 November 2015
How many subsets of two elements can be removed from the set so that the mean (average) of the remaining numbers is 6?
Solution
Since there will be elements after removal, and their mean is
, we know their sum is
. We also know that the sum of the set pre-removal is
. Thus, the sum of the
elements removed is
. There are only
subsets of
elements that sum to
:
. Therefore, our answer is
.