Elementary Foundations: An introduction to topics in discrete mathematics

You are trying to decide how to top your ice-cream sundae. You have five choices of sprinkles, four choices of cookie crumbs, five choices of fruit, and three choices of chocolate chunks. For each category of topping, you may choose only one of the available options, or you may choose to skip that category altogether. How many different sundaes could you create out of these choices?

You turn eighteen and your trust fund finally starts paying out. You decide to buy a vehicle, and eventually narrow things down to a choice between five SUVs, four sports cars, and two motorcycles. How many ways are there to choose a vehicle? How many ways are there to choose one vehicle of each type?

Recall that if $A$ is a finite set with $\left|A\right|=n\text{,}$ then $|\mathcal{P}(A)|=2^n\text{.}$ Use the Multiplication Rule to verify this formula by considering the construction of an arbitrary subset of $A$ as a process of making $n$ “either-or” decisions.