Problem 2 · AMC 8 Stretch
Core
Counting & Probability
and-process-multiplyaccounting-for-all-possibilities
A pizza shop lets you start with plain cheese and then add any of these 6 toppings: peppers, onions, sausage, mushrooms, broccoli, anchovies. You may add none, some, or all of them. If you order a different pizza every day, how many days until you have tried every possible pizza?
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Answer: 64 days
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Hint 1 of 4
There is no word 'and' in the problem, but an AND process is hiding. Go through the toppings one at a time.
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Hint 2 of 4
For each single topping you make one decision: yes (add it) or no (skip it). That is 2 choices per topping.
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Hint 3 of 4
You decide about topping 1 AND topping 2 AND ... AND topping 6 — six yes/no decisions in a row. Multiply the choices.
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Approach: AND process — a yes/no choice for each topping
- Walk through the toppings one at a time and ask: do I add this one? Each answer is Yes or No — that is 2 choices — and you do it for all 6 toppings.
- Since you decide topping 1 AND topping 2 AND ... AND topping 6, you multiply: \(2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^6 = 64\).
- The 'No to all six' outcome is just the plain cheese slice, so it is correctly included. There are 64 different pizzas, so it takes 64 days.
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