Mario orders a pizza with 3 toppings, chosen from 15 available toppings.
In how many ways could mushrooms OR olives be included in his toppings?
To find the number of ways mushrooms OR olives can be included in Mario's toppings, we need to calculate the total number of ways mushrooms can be included, the total number of ways olives can be included, and then subtract the cases where both mushrooms and olives are included (to avoid double counting).
1. Calculate the total number of ways mushrooms can be included:
Since mushrooms can be included or not included, there are 2 possibilities for each topping choice (mushrooms included or mushrooms not included). Since Mario can choose 3 toppings in total, the number of ways mushrooms can be included is given by 2^3 = 8.
2. Calculate the total number of ways olives can be included:
Similarly, since olives can be included or not included, there are also 2 possibilities for each topping choice. Therefore, the number of ways olives can be included is 2^3 = 8.
3. Calculate the number of ways both mushrooms and olives are included:
Since mushrooms and olives cannot be included together (as per the question's condition), the number of ways both mushrooms and olives can be included is 0.
4. Subtract the cases where both mushrooms and olives are included:
To avoid double counting, we need to subtract the cases where both mushrooms and olives are included. In this case, there are 0 such possibilities.
5. Calculate the final answer:
To find the number of ways mushrooms OR olives can be included, we add the number of ways mushrooms can be included (8) with the number of ways olives can be included (8), and then subtract the cases where both mushrooms and olives are included (0).
Therefore, the final answer is 8 + 8 - 0 = 16.