Difference between revisions of "2010 AIME II Problems/Problem 8"
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== Problem == | == Problem == | ||
Let <math>N</math> be the number of [[ordered pair]]s of nonempty sets <math>\mathcal{A}</math> and <math>\mathcal{B}</math> that have the following properties: | Let <math>N</math> be the number of [[ordered pair]]s of nonempty sets <math>\mathcal{A}</math> and <math>\mathcal{B}</math> that have the following properties: |
Revision as of 17:03, 9 August 2018
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 .
See also
2010 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.