Difference between revisions of "2006 AMC 12B Problems/Problem 9"
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− | 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 | + | 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 7, 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 2, there is 1 possibility for the hundreds digit, 3 for the ones digit. |
Revision as of 15:20, 12 September 2019
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 7, 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 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 .
Solution 3
The answer must be half of a triangular number (evens and decreasing/increasing) so or B. -zoevv
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.