There are 42 runners in a race. How many ways can the runners finish​ first, second, and​ third?
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Part 1
There are 
  
enter your response here different ways that the runners can finish first through third.
​(Type a whole​ number.)

To solve this problem, we need to find the number of ways the runners can finish in the top three positions, considering that the order of finishing matters (i.e., who finishes first, second, and third).

Since there are 42 runners in the race, the first place can be filled by any of the 42 runners. After the first place is filled, only 41 runners remain for the second place, and after the second place is filled, 40 runners remain for the third place.

Thus, the total number of ways to assign first, second, and third places is:

\[
42 \times 41 \times 40 = 68,760
\]

So, there are **68,760** different ways the runners can finish first, second, and third.

Would you like further explanation or assistance with a related question?

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Here are 5 related questions for practice:
1. How many different ways can 5 runners finish first, second, and third?
2. If there are 10 runners in a race, how many ways can the top 3 finishers be arranged?
3. How many ways can the first, second, and third places be assigned in a race with 50 runners?
4. If a race has 42 runners, how many ways can the top 4 places be filled?
5. How does the formula change if we were interested in finding the number of ways to select top 3 runners without regard to order?

**Tip:** The number of arrangements of a subset of items where order matters is calculated using permutations, and the formula for this is \( P(n, r) = \frac{n!}{(n - r)!} \), where \( n \) is the total number of items, and \( r \) is the number of items being selected.

There are 42 runners in a race. How many ways can the runners finish first, second, and third?

Learn how to calculate the number of different ways the top three finishers can be arranged in a race with 42 runners. This problem involves permutations and provides a clear solution using the permutation formula. Suitable for students in Grades 7-9.

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