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The problem asks for the number of different possible sets of winners in a race where there are **10 runners**, and awards are given for **1st, 2nd, and 3rd** places.

This is a **permutation** problem because the order in which the winners are selected matters (1st, 2nd, and 3rd place are distinct).

### Step 1: Identify the Correct Formula
For a permutation problem where the number of items to choose from is \(n\), and the number of positions to fill is \(r\), the correct formula is:

\[
\frac{n!}{(n - r)!}
\]

So the formula you should circle is:

\[
\frac{n!}{(n - r)!}
\]

### Step 2: Fill in the Values
In this case:
- \(n = 10\) (since there are 10 runners)
- \(r = 3\) (since 3 places are awarded)

Substitute these values into the formula:

\[
\frac{10!}{(10 - 3)!} = \frac{10!}{7!}
\]

### Step 3: Simplify and Compute
Now, we can compute this expression by canceling out the common factorial terms:

\[
\frac{10!}{7!} = 10 \times 9 \times 8
\]

\[
10 \times 9 = 90
\]
\[
90 \times 8 = 720
\]

Thus, the total number of different possible sets of winners is **720**.

---

Would you like more details or have any questions?

Here are 5 related questions to explore:
1. How would the solution change if there were only awards for 1st and 2nd place?
2. What if there were awards for 4 places instead of 3?
3. How would you calculate the number of possible sets if the order didn’t matter (a combination instead of a permutation)?
4. What happens if you have more runners, say 12, but still award 3 places?
5. Can this formula be applied to non-race situations, like assigning people to ranked positions in a committee?

**Tip:** When determining whether to use permutations or combinations, remember that **order matters** for permutations, but **not for combinations**.

If 10 runners have a race and there are awards for 1st, 2nd, and 3rd place, how many different possible sets of winners are there?

Learn how to solve a permutation problem involving 10 runners in a race and awards for 1st, 2nd, and 3rd place. This detailed step-by-step solution walks through using the formula n! / (n - r)! to calculate 720 possible sets of winners. Suitable for high school students in Grades 9-12.

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