64 Athletes are running a race. A gold medal is to be given to the winner, a silver medal is to be given to the second-place finisher, and a bronze medal is to be given to the third-place finisher. Assume that there are no ties. In how many possible ways can the 3 medals be distributed?

To determine how many possible ways the 3 medals can be distributed among the 64 athletes, we need to calculate the number of different ways to assign the gold, silver, and bronze medals.

Since there are no ties, and each medal goes to a different athlete, this is a permutation problem.

### Step-by-step calculation:
1. **Choosing the gold medalist**: There are 64 athletes available, so we have 64 choices for the gold medal.
   
2. **Choosing the silver medalist**: After the gold medalist is chosen, 63 athletes remain. Therefore, there are 63 choices for the silver medal.
   
3. **Choosing the bronze medalist**: After the gold and silver medalists are chosen, 62 athletes remain. Hence, there are 62 choices for the bronze medal.

To find the total number of ways to distribute the medals, we multiply these choices together:

\[
64 \times 63 \times 62 = 249,984
\]

Thus, the number of possible ways to distribute the gold, silver, and bronze medals is **249,984**.

Would you like further details or have any other questions?

Here are 5 related questions:
1. How many ways can the top 4 positions in the race be arranged?
2. What if there were only 10 athletes? How many ways can the medals be distributed?
3. If ties were allowed, how would the number of distributions change?
4. How many different ways can the top 3 be arranged if only the silver and bronze medals are considered?
5. What if the athletes are divided into teams of 4 and medals are awarded to teams instead of individuals?

**Tip**: When calculating permutations, remember the difference between selecting items with and without replacement, as this affects how the choices are counted.

Explore how to calculate the number of ways to distribute gold, silver, and bronze medals among 64 athletes using permutations. Understand the step-by-step process and the formula involved. Suitable for high school students, Grades 10-12.

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