2004 AMC 10A Problem 12

A hamburger is decided by two separate choices: the condiments and the number of patties.

1. Condiments. Go through the $8$ condiments one at a time and decide yes or no for each. That is $2$ choices, $8$ times:
$$2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^8 = 256.$$
This includes the plain burger with no condiments, which "any collection" allows.
2. Patties. One, two or three: $3$ choices.
3. Every condiment choice can go with every patty choice, so multiply:
$$3 \times 256 = 768.$$

The answer is $\boxed{\textbf{(C)}\ 768}$.

"Any collection" of $n$ items means you decide in or out for each one, which gives $2^n$ possibilities. Separate choices multiply. The order in which condiments go on the burger is not being asked about, so factorials do not belong here.

A hamburger is specified by two independent decisions.

Condiments: for each of the $8$ condiments the customer decides yes or no, so there are $2^8 = 256$ possible collections (including the empty one).

Patties: $3$ options.

By the multiplication principle the number of hamburgers is $3 \cdot 256 = 768$.

The answer is $\boxed{\textbf{(C)}\ 768}$.