2012 AMC 12B Problems/Problem 18
Problem 18
Let be a list of the first 10 positive integers such that for each either or or both appear somewhere before in the list. How many such lists are there?
Solution
Let . Assume that . If , the first number appear after that is greater than must be , otherwise if it is any number larger than , there will be neither nor appearing before . Similarly, one can conclude that if , the first number appear after that is larger than must be , and so forth.
On the other hand, if , the first number appear after that is less than must be , and then , and so forth.
To count the number of possibilities when is given, we set up spots after , and assign of them to the numbers less than and the rest to the numbers greater than . The number of ways in doing so is choose .
Therefore, when summing up the cases from to , we get
See Also
2012 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 17 |
Followed by Problem 19 |
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