Seven athletes are participating in a race. In how many ways can the first three athletes win the prizes?

Answer

To solve the problem of how many ways the first three athletes can win the prizes among seven athletes, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Total Number of Athletes**:
We have a total of 7 athletes participating in the race.

2. **Determine the Number of Prizes**:
We need to find the number of ways to award the first three prizes (1st, 2nd, and 3rd).

3. **Choose the First Prize Winner**:
For the first prize, any of the 7 athletes can win. So, there are 7 possible choices for the first prize.

4. **Choose the Second Prize Winner**:
After the first prize has been awarded, only 6 athletes remain eligible for the second prize. Thus, there are 6 possible choices for the second prize.

5. **Choose the Third Prize Winner**:
After awarding the first and second prizes, only 5 athletes remain for the third prize. Therefore, there are 5 possible choices for the third prize.

6. **Calculate the Total Number of Ways**:
To find the total number of ways to award the first three prizes, we multiply the number of choices for each prize:
\[ \text{Total Ways} = 7 \times 6 \times 5 \]

7. **Perform the Calculation**:
\[ 7 \times 6 = 42 \]
\[ 42 \times 5 = 210 \]

8. **Final Answer**:
The total number of ways the first three athletes can win the prizes is **210**.