Question

In how many ways can 8 athletes be arranged to start a race if a running track has eight lanes?

Answer 1

The result is `8!`, which is `8 \cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2=40320`. This is, in general, the number of permutations of 8 elements (in your case, the permutations of the 8 runners arranged in the 8 lanes). To see it, you can solve the problem this way: you have 8 possible choices for the first lane, since no runner has been placed on any lane yet. Now, you must choose which runner you place on the second lane, and you have 7 choices, since you already selected a runner for the first lane, and so on.