QQuestionMathematics
QuestionMathematics
"At a competition with 6 runners, 6 medals are awarded for first place through sixth place. Each medal is different.
How many ways are there to award the medals? Decide if the situation involves a permutation or a combination, and then find the number of ways to award the medals."
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Answer
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Step 1:I'll solve this problem step by step, following the specified LaTeX formatting guidelines:
Step 2:: Identify the Problem Type
This is a permutation problem because: - The order matters (first place is different from second place) - All 6 medals are distinct - We are arranging all 6 runners in a specific order
Step 3:: Determine the Calculation Method
Where $$n$$ is the total number of runners/medals
For permutations of all items, we use the factorial formula:
Step 4:: Calculate the Number of Permutations
Number of ways to award medals = $$6!
Step 5:: Compute the Factorial
6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1
6! = 720
Final Answer
Key Insights: - Permutations are used when order matters - The factorial calculates all possible arrangements - Each runner can receive only one medal - The medals are distinct (different for each place)
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