Difference between revisions of "2003 AIME I Problems/Problem 3"
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== Problem == | == Problem == | ||
| − | Let the set <math> \mathcal{S} = \{8, 5, 1, 13, 34, 3, 21, 2\}. </math> Susan makes a list as follows: for each two-element subset of <math> \mathcal{S}, </math> she writes on her list the greater of the set's two elements. Find the sum of the numbers on the list. | + | Let the [[set]] <math> \mathcal{S} = \{8, 5, 1, 13, 34, 3, 21, 2\}. </math> Susan makes a list as follows: for each two-element subset of <math> \mathcal{S}, </math> she writes on her list the greater of the set's two elements. Find the sum of the numbers on the list. |
== Solution == | == Solution == | ||
Each [[element]] of the [[set]] will appear in <math>7</math> two-element [[subset]]s, once with each other number. | Each [[element]] of the [[set]] will appear in <math>7</math> two-element [[subset]]s, once with each other number. | ||
| − | <math>34</math> will be the greater number in <math>7</math> subsets. | + | *<math>34</math> will be the greater number in <math>7</math> subsets. |
| + | *<math>21</math> will be the greater number in <math>6</math> subsets. | ||
| + | *<math>13</math> will be the greater number in <math>5</math> subsets. | ||
| + | *<math>8</math> will be the greater number in <math>4</math> subsets. | ||
| + | *<math>5</math> will be the greater number in <math>3</math> subsets. | ||
| + | *<math>3</math> will be the greater number in <math>2</math> subsets. | ||
| + | *<math>2</math> will be the greater number in <math>1</math> subsets. | ||
| + | *<math>1</math> will be the greater number in <math>0</math> subsets. | ||
| − | <math>21</math> | + | Therefore the desired sum is <math>34\cdot7+21\cdot6+13\cdot5+8\cdot4+5\cdot3+3 \cdot2+2\cdot1+1\cdot0=\boxed{484}</math>. |
| − | + | == See also == | |
| + | {{AIME box|year=2003|n=I|num-b=2|num-a=4}} | ||
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[[Category:Introductory Combinatorics Problems]] | [[Category:Introductory Combinatorics Problems]] | ||
Revision as of 13:59, 10 June 2008
Problem
Let the set 
Susan makes a list as follows: for each two-element subset of 
she writes on her list the greater of the set's two elements. Find the sum of the numbers on the list.
Solution
Each element of the set will appear in 
two-element subsets, once with each other number.
will be the greater number in 
subsets.
will be the greater number in 
subsets.
will be the greater number in 
subsets.
will be the greater number in 
subsets.- will be the greater number in
subsets. - will be the greater number in
subsets. - will be the greater number in
subsets. - will be the greater number in
subsets.
will be the greater number in 
will be the greater number in 
subsets.
will be the greater number in 
subsets.
will be the greater number in 
subsets.
subsets.
subsets.
subsets.
subsets.Therefore the desired sum is 
.
See also
| 2003 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 2 |
Followed by Problem 4 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
