A sandwich with optional toppings

Question

At a deli you build a sandwich by choosing $1$ of $4$ breads and $1$ of $3$ proteins. Then, for each of $3$ toppings (lettuce, tomato, onion), you independently decide to add it or skip it.

How many different sandwiches are possible?

Accepted Answer

Break the sandwich into independent decisions and multiply. - Bread: 4 ways. - Protein: 3 ways. - Each topping is a separate yes/no choice, so it contributes a factor of 2. With 3 toppings that is 2 	imes 2 	imes 2 = 2^3 = 8. Multiplying all stages: 4 	imes 3 	imes 2^3 = 4 	imes 3 	imes 8 = 96. The twist here is that an "add it or skip it" option is itself a choice with 2 outcomes, so each optional topping doubles the total. Takeaway. A yes/no option multiplies the count by 2. With k independent optional add-ons, the totals get a factor of 2^k: here 4 	imes 3 	imes 2^3 = 96.

A sandwich with optional toppings

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