1997 AIME Problems/Problem 10
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?
Contents
Solution 2
Let's say we have picked two cards. We now compare their attributes to decide how we can pick the third card to make a complement set. For each of the three attributes, should the two values be the same we have one option - choose a card with the same value for that attribute. Furthermore, should the two be different there is only one option- choose the only value that is remaining. In this way, every two card pick corresponds to exactly one set, for a total of 
possibilities. Note, however, that each set is generated by 
pairs, so we've overcounted by a multiple of 3 and the answer is 
.
Solution 3
Treat the sets as ordered. Then for each of the three criterion, there are 
choices if the attribute is different and there are 
choices is the attribute is the same. Thus all three attributes combine to a total of 
possibilities. However if all three attributes are the same then the set must be composed of three cards that are the same, which is impossible. This takes out 
possibilities. Notice that we have counted every set times by treating the set as ordered. the final solution is then 
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 | ||
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