2021 AIME II Problems/Problem 6
Problem
For any finite set , let
denote the number of elements in
. FInd the number of ordered pairs
such that
and
are (not necessarily distinct) subsets of
that satisfy
Solution 1
By PIE, , and after some algebra you see that we need
or
. WLOG
, then for each element there are
possibilities, either it is in both
and
, it is in
but not
, or it is in neither
nor
. This gives us
possibilities, and we multiply by
since it could of also been the other way around. Now we need to subtract the overlaps where
, and this case has
ways that could happen. It is
because each number could be in the subset or it could not be in the subset. So the final answer is
.
~ math31415926535
Solution 2
We denote .
We denote
,
,
,
.
Therefore, and the intersection of any two out of sets
,
,
,
is an empty set.
Therefore,
is a partition of
.
Following from our definition of ,
,
, we have
.
Therefore, the equation
can be equivalently written as
This equality can be simplified as
~ Steven Chen (www.professorchenedu.com)
2021 AIME II (Problems • Answer Key • Resources) | ||
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Followed by Problem 7 | |
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