# Problem of the Day

At Mike’s Hamburgers, each hamburger can be ordered with any of the following toppings: lettuce, tomato, onion, pickle, mayonnaise, mustard, ketchup, cheese. How many different combinations of toppings are possible?

A. 8

B. 56

C. 128

D. 256

E. 512

### Solution

[latexpage]

When a customer orders a hamburger, he has 8 decisions to make. Lettuce: yes or no? Tomato: yes or no?……. Cheese: yes or no? He has 8 decisions to make and each decision can be made in 2 ways. Therefore there are $2^8$ possible combinations of toppings for him to choose from.

If this is unclear to you, start with a smaller number of toppings and list the possibilities. For example, if the only possible topping was lettuce, there would be two possible ways to order the hamburger: with or without lettuce. If only lettuce and tomato were available then the customer could order from four possible hamburgers: lettuce and tomato, lettuce without tomato, tomato without lettuce, no lettuce and no tomato. So with 1 topping, the answer would be $2^1$ and with 2 toppings the answer would be $2^2$. In general, with $n$ toppings there are $2^n$ possible combinations of toppings.

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