Considering the outcome of runners, which calculation gives the total permutations of 3 from 7 runners?

To determine the total permutations of selecting 3 runners from a group of 7, we need to consider the order in which these runners are arranged. The correct calculation involves the concept of permutations, which takes into account all possible arrangements of the selected items.

When calculating permutations, the formula used is:

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

In this case, ( n ) is the total number of items (runners) we have, which is 7, and ( r ) is the number of items we want to arrange, which is 3. Substituting these values into the formula gives us:

[ P(7, 3) = \frac{7!}{(7 - 3)!} = \frac{7!}{4!} ]

This result indicates that we first calculate the factorial of 7, which represents all possible arrangements of the 7 runners, and then we divide by the factorial of 4, which accounts for the arrangements of the runners not selected.

The choice that reflects this calculation—dividing the factorial of the total number of runners by the factorial of the number of runners not selected—is the first option,