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To solve the problem, we need to find the number of ways the first three finishers can be arranged from six automobiles. This is a permutation problem, as the order of the finishers matters.

### Solution:
We have 6 automobiles and we want to choose and arrange the first 3 of them. The formula for permutations is:

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

where:
- \( n \) is the total number of items (6 automobiles),
- \( r \) is the number of items we want to arrange (3 finishers).

\[
P(6, 3) = \frac{6!}{(6-3)!} = \frac{6!}{3!}
\]

Calculating this:

\[
P(6, 3) = \frac{6 \times 5 \times 4 \times 3!}{3!} = 6 \times 5 \times 4 = 120
\]

Therefore, **there are 120 ways** for the first three finishers to come in.

Would you like more details or have any questions about the solution?

### Further Questions:
1. How would the answer change if there were 8 automobiles instead of 6?
2. What if we wanted to determine how many ways all 6 automobiles could finish?
3. Can you calculate the number of ways for the top 4 finishers instead of 3?
4. How would this change if there were ties among the automobiles?
5. What is the difference between permutations and combinations in this context?

**Tip:** In permutation problems, remember that order matters, unlike combinations where order does not matter.

In a race in which six automobiles are entered and there are no ties, in how many ways can the first three finishers come in?

Learn how to calculate the number of ways the first three finishers can be arranged in a race of six automobiles. This permutation problem involves the formula P(n, r) = n! / (n - r)! and covers key combinatorics concepts. Suitable for Grades 9-12.

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