at a competition with 6 runners

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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Introduction:

In a competition with six runners, six distinct medals are awarded for the first to sixth place positions. Each medal is unique, meaning that the runners are ranked differently for each medal they receive. The question asks for the number of ways these medals can be awarded.

Explanation:

This situation involves a permutation, not a combination. A permutation is used when the order of selection matters, and in this case, the order in which the runners are awarded medals is crucial. Each runner gets a specific medal based on their performance, and the medals are distinct, meaning the rankings for each place are important.

Answer:

To calculate the number of ways to award the medals, we use the formula for permutations, which is given by:

$$P(n, r) = \frac{n!}{(n - r)!}$$

In this case, we are awarding medals to all 6 runners, so $n = r = 6$. Therefore, the number of ways to award the medals is:

$$P(6, 6) = 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720$$

Thus, there are 720 ways to award the six distinct medals to the six runners.