Question

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:

• Either each of the three cards has a different shape or all three of the cards have the same shape.

• Either each of the three cards has a different color or all three of the cards have the same color.

• 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?

Answer

Solution:

Given any two distinct cards, there is exactly one card completing them to a complementary set: in each attribute, if the two cards agree, the third card must share that value, and if they differ, the third must take the one remaining value. The completing card is distinct from both (the two given cards differ somewhere, and in that attribute the third card differs from each).

So the $\binom{27}{2} = 351$ pairs of cards each extend to one complementary set, and each complementary set is produced by $\binom{3}{2} = 3$ of these pairs. The number of sets is $\frac{351}{3} = 117.$

1997 AIME Problem 10

Solution:

Problem 10 in Other Years