Assume that 15 exhibits entered the science fair, and that we want to assign 3 prizes: first, second, and third. How many different ways can the prizes be awarded?

A member of the Student Council decided to conduct a room visit and asked the students for their club involvement. Out of 150 who were surveyed, 85 mentioned that they joined the Debate Club; 70 said that they signed up for the Cooking Club; and 50 joined both the Debate and Cooking Clubs. How many students joined the Debate or Cooking Club?

A birthday cake is to have two layers of cake separated by a layer of icing, with another layer of icing on top. There are 3 kinds of cake the chef knows how to make, and she may use either two different kinds or the same one twice for the two cake layers. The chef knows recipes for 4 different icings, and she will use two different recipes, one for each icing layer, in this cake. How many different cakes could the chef make? Note that the order of the layers matters, e.g. chocolate over strawberry is different from strawberry over chocolate.

The Student Store holds a contest to hand out souvenir shirts featuring four designs from the Mathematics Hall of Fame. Each shirt is size XL. 700 different people enter the contest. 6 different people win a shirt. Shirts with the same design are indistinguishable from one another. The order in which the shirts are handed out does not matter. How many different ways can the shirts be handed out? No need to simplify your answer. You must explain your answer.