Ten students are participating in a race. In how many ways can the first three prizes be won?

Explanation

To find the number of ways the first three prizes can be won among ten students, we can use the concept of permutations since the order in which the prizes are awarded matters.

### Step-by-step Solution:

1. **Identify the total number of participants**:
There are 10 students participating in the race.

2. **Determine the number of prizes**:
We need to find the number of ways to award the first three prizes.

3. **Use the permutation formula**:
The number of ways to arrange \( r \) items from \( n \) items is given by the formula:
\[ P(n, r) = \frac{n!}{(n - r)!} \]
where \( n \) is the total number of items (students) and \( r \) is the number of items to arrange (prizes).

4. **Substitute the values into the formula**:
Here, \( n = 10 \) (students) and \( r = 3 \) (prizes).
\[ P(10, 3) = \frac{10!}{(10 - 3)!} = \frac{10!}{7!} \]

5. **Simplify the factorials**:
The factorial \( 10! \) can be expressed as:
\[ 10! = 10 \times 9 \times 8 \times 7! \]
Therefore,
\[ P(10, 3) = \frac{10 \times 9 \times 8 \times 7!}{7!} = 10 \times 9 \times 8 \]

6. **Calculate the result**:
Now, calculate \( 10 \times 9 \times 8 \):
\[ 10 \times 9 = 90 \]
\[ 90 \times 8 = 720 \]

7. **Conclusion**:
Thus, the total number of ways the first three prizes can be won is **720**.