Question 10
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?
• 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?