62 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?
Question
62 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?
Solution
The problem can be solved using the concept of permutations in combinatorics.
Step 1: Determine the number of ways to choose the gold medalist. There are 62 athletes, so there are 62 ways to choose the gold medalist.
Step 2: Determine the number of ways to choose the silver medalist. After the gold medalist has been chosen, there are 61 remaining athletes, so there are 61 ways to choose the silver medalist.
Step 3: Determine the number of ways to choose the bronze medalist. After the gold and silver medalists have been chosen, there are 60 remaining athletes, so there are 60 ways to choose the bronze medalist.
Step 4: Since each of these choices is independent, the total number of ways to distribute the medals is the product of the number of choices at each step. So, the total number of ways to distribute the medals is 62 * 61 * 60 = 226,920.
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