2010 AIME II Problems/Problem 8
Contents
Problem
Let be the number of ordered pairs of nonempty sets
and
that have the following properties:
-
,
-
,
- The number of elements of
is not an element of
,
- The number of elements of
is not an element of
.
Find .
Solution
Let us partition the set into
numbers in
and
numbers in
,
Since must be in
and
must be in
(
, we cannot partition into two sets of 6 because
needs to end up somewhere,
or
either).
We have ways of picking the numbers to be in
.
So the answer is .
Solution 2
Regardless of the size of
(ignoring the case when
),
must not be in
and
must be in
.
There are remaining elements who’s placements have yet to be determined. Note that the actual value of
does not matter; there is always
necessary element,
forbidden element, and
other elements that need to be distributed. There are
places to put each of these elements, for
possibilities.
However, this ignores the case when is forced not the be in either set, so we must subtract the
cases where
and
have size
.
Thus, our answer is
See also
2010 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
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