There are 8 sprinters in the Olympic 100-meter finals. Three of the sprinters are Americans.  The gold medal goes to first place, silver to second, and bronze to third.   In how many ways can the medals be awarded if at most one American gets a medal?
Consider two cases:

Case 1: No Americans get a medal. So there are 5 choices for the gold medal, 4 choices for the silver, and 3 choices for bronze, which is $5\times4\times3=60$ ways.

Case 2: One American gets a medal. There are 3 Americans to choose from. After we choose which American to get the medal, we have to decide which medal to reward the American, for which we have 3 choices. Then we have 5 choices for one of the remaining medal and 4 choices for the last medal. So, we have a total of $3\times3\times5\times4=180$ ways.

Sum up the two cases, and we have a total of $180+60=\boxed{240}$ ways.
240
Level 5
Prealgebra
Problem: There are 8 sprinters in the Olympic 100-meter finals. Three of the sprinters are Americans.  The gold medal goes to first place, silver to second, and bronze to third.   In how many ways can the medals be awarded if at most one American gets a medal?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the number of ways to award medals with at most one American sprinter receiving a medal, we need to consider two scenarios: no Americans winning medals and exactly one American winning a medal.

1. **Scenario 1: No Americans win medals.**
   - There are 5 non-American sprinters available.
   - We need to select 3 out of these 5 sprinters to receive the gold, silver, and bronze medals.
   - The number of ways to choose 3 sprinters from 5 is given by the combination formula C(n, k), where n is the total number of sprinters and k is the number of sprinters to choose. 
   - C(5, 3) = 10.
   - For each selection of 3 sprinters, they can be arranged in 3! (factorial of 3) ways to assign the medals.
   - 3! = 6.
   - Therefore, the total number of ways for this scenario is 10 * 6 = 60.

2. **Scenario 2: Exactly one American wins a medal.**
   - We have 3 American sprinters and 5 non-American sprinters.
   - First, we choose 1 American sprinter to win a medal. There are 3 ways to choose 1 American from 3.
   - Next, we need to select 2 non-American sprinters from the 5 available. The number of ways to choose 2 from 5 is C(5, 2) = 10.
   - For each selection of 1 American and 2 non-Americans, we can arrange these 3 sprinters in 3! ways to assign the medals.
   - Thus, the total number of ways for this scenario is 3 * 10 * 6 = 180.

3. **Final Calculation:**
   - We add the totals from both scenarios to find the overall number of ways to award the medals.
   - Total = 60 (no Americans) + 180 (one American) = 240.

The final answer is 240.<|END_OF_LATENT|> The final answer is $240$.