Difference between revisions of "2002 AMC 12B Problems/Problem 10"
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==Problem== | ==Problem== | ||
How many different integers can be expressed as the sum of three distinct members of the set <math>\{1,4,7,10,13,16,19\}</math>? | How many different integers can be expressed as the sum of three distinct members of the set <math>\{1,4,7,10,13,16,19\}</math>? | ||
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| − | ==Solution== | + | <math>\text{(A)}\ 13 \qquad \text{(B)}\ 16 \qquad \text{(C)}\ 24 \qquad \text{(D)}\ 30 \qquad \text{(E)}\ 35</math> |
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| + | ==Solution 1== | ||
| + | Subtracting 10 from each number in the set, and dividing the results by 3, we obtain the set <math>\{-3, -2, -1, 0, 1, 2, 3\}</math>. It is easy to see that we can get any integer between <math>-6</math> and <math>6</math> inclusive as the sum of three elements from this set, for the total of <math>\boxed{\mathrm{(A) } 13}</math> integers. | ||
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| + | ==Solution 2== | ||
| + | The set is an arithmetic sequence of numbers each <math>1</math> more than a multiple of <math>3</math>. Thus the sum of any three numbers will be a multiple of <math>3</math>. All the multiples of <math>3</math> from <math>1+4+7=12</math> to <math>13+16+19=48</math> are possible, totaling to <math>\boxed{\mathrm{(A) } 13}</math> integers. | ||
==See also== | ==See also== | ||
| + | {{AMC12 box|year=2002|ab=B|num-b=9|num-a=11}} | ||
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| + | [[Category:Introductory Combinatorics Problems]] | ||
| + | {{MAA Notice}} | ||
Latest revision as of 23:43, 16 January 2021
Contents
Problem
How many different integers can be expressed as the sum of three distinct members of the set 
?
Solution 1
Subtracting 10 from each number in the set, and dividing the results by 3, we obtain the set 
. It is easy to see that we can get any integer between 
and 
inclusive as the sum of three elements from this set, for the total of 
integers.
Solution 2
The set is an arithmetic sequence of numbers each 
more than a multiple of 
. Thus the sum of any three numbers will be a multiple of . All the multiples of from 
to 
are possible, totaling to integers.
See also
| 2002 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 9 |
Followed by Problem 11 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
