Browse · MATH
Printjmc
counting and probability senior
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
Case 1: All three attributes are the same. This is impossible since sets contain distinct cards. Case 2: Two of the three attributes are the same. There are ways to pick the two attributes in question. Then there are ways to pick the value of the first attribute, ways to pick the value of the second attribute, and way to arrange the positions of the third attribute, giving us ways. Case 3: One of the three attributes are the same. There are ways to pick the one attribute in question, and then ways to pick the value of that attribute. Then there are ways to arrange the positions of the next two attributes, giving us ways. Case 4: None of the three attributes are the same. We fix the order of the first attribute, and then there are ways to pick the ordering of the second attribute and ways to pick the ordering of the third attribute. This gives us ways. Adding the cases up, we get .
Final answer
117