Explanation

In this case, order matters, so you simply need to multiply the number of possible runners for each spot. That should look like this: \(12 \times 11 \times 10 = 1,320\).
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ASIDE: I'm somewhat surprised that this official practice question is somewhat ambiguous. It suggests that the " winners" are the runners who finish first second or third.

Take the task of listing possible winners and break it into stages.

Stage 1: Select a runner to be first
Since there are 12 runners to choose from, we can complete stage 1 in 12 ways

Stage 2: Select a runner to be second
Since we already chose a first place runner in stage 1, there are 11 runners REMAINING to choose from.
So we can complete stage 2 in 11 ways

Stage 3: Select a runner to be 3rd
There are now 10 runners remaining, so we can complete stage 3 in 10 ways

By the Fundamental Counting Principle (FCP), we can complete all 3 stages (list the arrangements of winners) in (12)(11)(10) ways (= 1320 ways)

Answer: 1320

Note: the FCP can be used to solve the MAJORITY of counting questions on the GRE. So, be sure to learn it.

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