Difference between revisions of "2006 AMC 12B Problems/Problem 9"
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== Solution == | == Solution == | ||
+ | ===Solution 1=== | ||
Let's set the middle (tens) digit first. The middle digit can be anything from 2-7 (If it was 1 we would have the hundreds digit to be 0, if it was more than 8, the ones digit cannot be even). | Let's set the middle (tens) digit first. The middle digit can be anything from 2-7 (If it was 1 we would have the hundreds digit to be 0, if it was more than 8, the ones digit cannot be even). | ||
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So, the answer is <math>3(1+2)+2(3+4)+1(5+6)=\boxed{34} \Rightarrow B</math>. | So, the answer is <math>3(1+2)+2(3+4)+1(5+6)=\boxed{34} \Rightarrow B</math>. | ||
+ | |||
+ | ===Solution 2=== | ||
+ | The last digit is 2, 4, 6, or 8. | ||
+ | |||
+ | If the last digit is <math>x</math>, the possibilities for the first two digits correspond to 2-element subsets of <math>\{1,2,\dots,x-1\}</math>. | ||
+ | |||
+ | Thus the answer is <math>{1\choose 2} + {3\choose 2} + {5\choose 2} + {7\choose 2} = 0 + 3 + 10 + 21 = 34</math>. | ||
== See also == | == See also == | ||
{{AMC12 box|year=2006|ab=B|num-b=8|num-a=10}} | {{AMC12 box|year=2006|ab=B|num-b=8|num-a=10}} |
Revision as of 19:30, 5 January 2009
Problem
How many even three-digit integers have the property that their digits, read left to right, are in strictly increasing order?
Solution
Solution 1
Let's set the middle (tens) digit first. The middle digit can be anything from 2-7 (If it was 1 we would have the hundreds digit to be 0, if it was more than 8, the ones digit cannot be even).
If it was 2, there is 1 possibility for the hundreds digit, 3 for the ones digit. If it was 3, there are 2 possibilities for the hundreds digit, 3 for the ones digit. If it was 4, there are 3 possibilities for the hundreds digit, and 2 for the ones digit,
and so on.
So, the answer is .
Solution 2
The last digit is 2, 4, 6, or 8.
If the last digit is , the possibilities for the first two digits correspond to 2-element subsets of .
Thus the answer is .
See also
2006 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 8 |
Followed by Problem 10 |
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 |