If there are 7 runners, which mathematical concept is used to calculate the different ways to arrange the top 3 runners?

To determine the different ways to arrange the top 3 runners out of 7, permutations is the appropriate mathematical concept to apply. Permutations are used when the order of selection is important, which is the case when ranking or arranging items—in this instance, arranging runners.

In this situation, you can think of it as selecting one runner for the top position, another for second place, and a third for third place. The number of arrangements changes based on the chosen runner for each position, reflecting the significance of the order.

Combinations, on the other hand, would be relevant if the order didn't matter, such as simply selecting a group of runners without regard to their finishing position. Factorials provide a foundational mathematical concept useful for calculating permutations and combinations but do not directly answer the question about arrangements. Exponents relate to multiplication of like numbers but do not apply to this specific context of ranking. Thus, permutations is the correct way to explore the various arrangements of the top 3 runners from a group of 7.