In a horse​ race, how many different finishes among the first 3 places are possible if 8 horses are​ running? (Exclude​ ties)
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Part 1
There are a possible 
  
enter your response here different finishes.

To determine the number of different finishes among the first 3 places in a horse race with 8 horses running, we need to consider the permutations of 8 horses taken 3 at a time. This is a permutation problem because the order in which the horses finish matters.

The formula for permutations of \( n \) items taken \( r \) at a time is given by:

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

Here, \( n = 8 \) and \( r = 3 \).

So, we calculate:

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

Breaking it down:

\[ 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \]

\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 \]

Thus:

\[ P(8, 3) = \frac{8 \times 7 \times 6 \times 5!}{5!} = 8 \times 7 \times 6 = 336 \]

So, there are a possible **336 different finishes** among the first 3 places in a horse race with 8 horses running.

Would you like more details or have any questions?

Here are 5 questions you might want to ask:
1. How do you calculate combinations instead of permutations?
2. Can you explain factorial notation and how it is used in permutations?
3. What is the difference between permutations and combinations?
4. How would the calculation change if there were 10 horses instead of 8?
5. Can you give an example of another problem where permutations are used?

**Tip:** Always verify whether a problem requires permutations (order matters) or combinations (order doesn't matter) before solving.

Explore how to calculate the number of different finishes among the first 3 places in a horse race with 8 horses running. Learn the permutations formula and step-by-step calculation. Suitable for high school students studying permutations.

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