The problem asks for the number of different finishes among the first three places in a horse race with 10 horses running, excluding ties.

Solution:

This is a permutation problem because the order of finish matters.

The formula for permutations is: $P(n, r) = \frac{n!}{(n - r)!}$ where:

• $n$ is the total number of items (horses),
• $r$ is the number of positions to fill (top 3 places).

Here:

• $n = 10$,
• $r = 3$.

So: $P(10, 3) = \frac{10!}{(10 - 3)!} = \frac{10 \times 9 \times 8 \times 7!}{7!} = 10 \times 9 \times 8 = 720$

Final Answer:

There are 720 different finishes possible.

Would you like me to break this down further or explore related concepts?

Follow-up Questions:

1. What would the result be if the number of horses increased to 12 but the top 3 places were still considered?
2. How would the outcome change if ties were allowed in the top 3 places?
3. Can you calculate the total number of ways the top 5 places could be arranged from 10 horses?
4. How is the formula for permutations different from combinations, and when should each be used?
5. What real-world situations other than horse races use permutations?

Tip:

When solving permutation problems, always ensure the order of selection is relevant—if not, consider using combinations instead!