2004 AMC 10A Problem 12

Below is the professionally curated solution for Problem 12 of the 2004 AMC 10A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2004 AMC 10A solutions, or check the answer key.

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Concepts:multiplication principle subsets

Difficulty rating: 1190

12.

Henry's Hamburger Heaven offers its hamburgers with the following condiments: ketchup, mustard, mayonnaise, tomato, lettuce, pickles, cheese, and onions. A customer can choose one, two, or three meat patties, and any collection of condiments. How many different kinds of hamburgers can be ordered?

$24$

$256$

$768$

$40{,}320$

$120{,}960$

Solution:

Each of the $8$ condiments is independently in or out, giving $2^8 = 256$ condiment combinations.

For each of these there are $3$ choices of patty count, so the number of hamburgers is $3 \times 256 = 768.$

Thus, the correct answer is C.

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