How many different varieties of pizza can be made if you have the following choice: small medium, large; thin, hand tossed, pan; and $12$ toppings (cheese is an automatic), from which you may select from $0-12$.
What I did for this was $3C1 \cdot 3C1 \cdot 13C1$. However, I know my answer is incorrect. Is it because $13C1$ is not the proper was of expressing the different ways toppings could be chosen?
Best Answer
You made your error when you concluded that there were $13$ ways of selecting $0 - 12$ toppings. It is the subset of toppings that are included rather than the number of toppings that matters. For instance, knowing that two toppings have been selected does not tell you whether those toppings are mushrooms and olives or pepperoni and anchovies.
You are correct that there are three ways of selecting the size and three ways of selecting the way the pizza is prepared. Since you can select any subset of the $12$ toppings, including none, the number of varieties of pizza is $$\binom{3}{1}\binom{3}{1}\cdot 2^{12}$$ The reason the number of subsets of $12$ toppings is $2^{12}$ is that there are two options for each topping, to include it or not to include it.