\( 1 \leftarrow \quad \) There are 60 runners in a race. How many ways can the runners finish first, second, and third?

To determine the number of ways the runners can finish first, second, and third, we can think of this as a permutation problem where we want to arrange 3 out of 60 runners. The number of ways to choose 3 runners from 60, and arrange them in order, is given by the formula for permutations: \[ P(n, r) = \frac{n!}{(n - r)!} \] In this case, \(n = 60\) (the total number of runners) and \(r = 3\) (the positions we care about: first, second, and third). So, we calculate: \[ P(60, 3) = \frac{60!}{(60 - 3)!} = \frac{60!}{57!} = 60 \times 59 \times 58 \] Now let's perform the multiplication: \[ 60 \times 59 = 3540 \] \[ 3540 \times 58 = 205320 \] Thus, the number of ways the runners can finish in the top three positions is \(205320\).