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- In a race, there are 8 participants. In how many ways can the gold, silver, and bronze medals be awarded if the previous year's winner cannot win the gold medal again
In a race, there are 8 participants. In how many ways can the gold, silver, and bronze medals be awarded if the previous year's winner cannot win the gold medal again?
126 ways
420 ways
336 ways
720 ways
Answers (1)
To solve this, we need to consider the different cases based on who wins the gold medal:
Case 1: If the previous year's winner does not win the gold medal:
There are 7 participants who can win the gold medal.
Once the gold medalist is determined, there are 7 remaining participants who can win the silver medal.
After the silver medalist is determined, there are 6 remaining participants who can win the bronze medal.
Therefore, the number of ways for this case is: 7 choices for the gold medal × 7 choices for the silver medal × 6 choices for the bronze medal = 294 ways.
Case 2: If the previous year's winner wins the gold medal again:
There is 1 participant who can win the gold medal (the previous year's winner).
Once the gold medalist is determined, there are 7 remaining participants who can win the silver medal.
After the silver medalist is determined, there are 6 remaining participants who can win the bronze medal.
Therefore, the number of ways for this case is: 1 choice for the gold medal × 7 choices for the silver medal × 6 choices for the bronze medal = 42 ways.
To find the total number of ways, we sum the possibilities for each case:
Total number of ways = 294 ways + 42 ways = 336 ways.
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