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

Options

A. 920
B. 680
C. 820
D. 720

Answer

To solve the problem of how many ways the first three prizes can be won by ten students, we will 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 (n)**:
- There are 10 students participating in the race.
- So, \( n = 10 \).

2. **Identify the Number of Prizes (r)**:
- We are interested in the first three prizes.
- Thus, \( r = 3 \).

3. **Use the Permutation Formula**:
- The number of ways to arrange \( r \) items from \( n \) items is given by the formula:
\[ P(n, r) = \frac{n!}{(n - r)!} \]
- In our case, we need to calculate \( P(10, 3) \).

4. **Substitute the Values into the Formula**:
- Plugging in the values, we get:
\[ P(10, 3) = \frac{10!}{(10 - 3)!} = \frac{10!}{7!} \]

5. **Simplify the Factorials**:
- We can expand \( 10! \) as follows:
\[ 10! = 10 \times 9 \times 8 \times 7! \]
- Therefore:
\[ P(10, 3) = \frac{10 \times 9 \times 8 \times 7!}{7!} \]
- The \( 7! \) in the numerator and denominator cancels out:
\[ P(10, 3) = 10 \times 9 \times 8 \]

6. **Calculate the Result**:
- Now, we can calculate:
\[ 10 \times 9 = 90 \]
\[ 90 \times 8 = 720 \]

7. **Final Answer**:
- Thus, the total number of ways the first three prizes can be won is \( 720 \).