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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