A race has 8 runners and the first three finishers receive a prize. How many possible ways can the prize be distributed?

In this problem, the number of ways the prizes can be distributed depends on the order in which the runners finish, as the first, second, and third places are distinct.

Since there are 8 runners and the order matters, this is a permutation problem. The number of ways to select 3 winners from 8 runners, where order matters, is calculated by using the permutation formula:

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

Where:
- \( n \) is the total number of items (8 runners),
- \( r \) is the number of items to arrange (3 finishers),
- \( n! \) is the factorial of \( n \).

Let's calculate \( P(8, 3) \):

\[
P(8, 3) = \frac{8!}{(8 - 3)!} = \frac{8!}{5!}
\]

We know that:

\[
8! = 8 \times 7 \times 6 \times 5! = 8 \times 7 \times 6
\]

So:

\[
P(8, 3) = 8 \times 7 \times 6 = 336
\]

Thus, there are **336 possible ways** to distribute the prizes among the top three finishers.

Would you like a detailed breakdown of this process or have any other questions?

### Related Questions:
1. What if there were 10 runners and still only 3 prizes?
2. How would the solution change if the order of finishers did not matter?
3. What is the probability that a specific runner finishes in first place?
4. How many ways can prizes be distributed if there are 4 prizes instead of 3?
5. How would the problem change if 8 runners are grouped into teams and only team order matters?

### Tip:
In permutation problems, the order in which items are arranged is crucial. If order doesn't matter, the problem turns into a combination problem instead.

Learn how to calculate the number of ways to distribute prizes in a race with 8 runners, where the top 3 finishers receive prizes. This combinatorics problem uses the permutation formula, P(n, r), and involves calculating factorials. Suitable for students in Grades 9-12.

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