Cards in a Deck

Probability Level 4

Every card in a deck has a picture of one shape - circle, square, or triangle, which is painted in one of 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

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

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

3) 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?

The answer is 117.

1 solution

Alan Yan
Oct 8, 2015

Note that for every pair of cards, there is exactly one other card that makes it a complementary triplet. Then we need to divide by three for overcounting since each element of a triplet can be the "complement".

Therefore the answer is

${27 \choose 2} \cdot \frac{1}{3} = \boxed{117}$

Cards in a Deck

The answer is 117.

1 solution

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