Difference between revisions of "1997 AIME Problems/Problem 10"
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== Solution == | == Solution == | ||
| − | {{solution}} | + | We call these three types of complementary sets <math>A,B,C</math> respectively. What we are trying to find is |
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| + | <cmath>n(A\cup B\cup C)</cmath> | ||
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| + | We know this is equivalent to | ||
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| + | <cmath>n(A)+n(B)+n(C)-n(A\cap B)-n(B\cap C)-n(A\cap C)+n(A\cap B \cap C)</cmath> | ||
| + | |||
| + | Now, <math>n(A)=\binom{9}{3}+9^3=813</math>. Obviously, <math>n(B)</math> and <math>n(C)</math> are the same. Thus, we have | ||
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| + | <cmath>2439-n(A\cap B)-n(B\cap C)-n(A\cap C)+n(A\cap B \cap C)</cmath> | ||
| + | |||
| + | {{incomplete|solution}} | ||
== See also == | == See also == | ||
{{AIME box|year=1997|num-b=9|num-a=11}} | {{AIME box|year=1997|num-b=9|num-a=11}} | ||
Revision as of 21:57, 20 November 2007
Problem
Every card in a deck has a picture of one shape - circle, square, or triangle, which is painted in one of the three colors - red, blue, or green. Furthermore, each color is applied in one of three shades - light, medium, or dark. The deck has 27 cards, with every shape-color-shade combination represented. A set of three cards from the deck is called complementary if all of the following statements are true:
i. Either each of the three cards has a different shape or all three of the card have the same shape.
ii. Either each of the three cards has a different color or all three of the cards have the same color.
iii. Either each of the three cards has a different shade or all three of the cards have the same shade.
How many different complementary three-card sets are there?
Solution
We call these three types of complementary sets 
respectively. What we are trying to find is
![\[n(A\cup B\cup C)\]](http://latex.artofproblemsolving.com/6/0/5/6057a39271ca19203fedd2534662a26b14ee9c33.png)
We know this is equivalent to
![\[n(A)+n(B)+n(C)-n(A\cap B)-n(B\cap C)-n(A\cap C)+n(A\cap B \cap C)\]](http://latex.artofproblemsolving.com/e/f/8/ef87f46cf92a04ee5363ac49ab7f3585e7dd3ba6.png)
Now, 
. Obviously, 
and 
are the same. Thus, we have
![\[2439-n(A\cap B)-n(B\cap C)-n(A\cap C)+n(A\cap B \cap C)\]](http://latex.artofproblemsolving.com/3/a/7/3a70800e2ec1c8c28b95fcdf4b034907268c67c9.png)
See also
| 1997 AIME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 9 |
Followed by Problem 11 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
