How can the total arrangements of 1st, 2nd, and 3rd place in a race be expressed mathematically?

The total arrangements of 1st, 2nd, and 3rd place in a race can be expressed mathematically using the concept of permutations. When arranging a subset of items from a larger set, the order in which the items are arranged matters, which is the case for positions in a race.

In this scenario, you want to find the number of ways to select and arrange 3 positions (1st, 2nd, and 3rd) from a total of 7 participants. This can be mathematically represented as the permutation of 7 items taken 3 at a time, typically denoted as (7P3). The formula for permutations is given by:

[

nPr = \frac{n!}{(n-r)!}

]

In this case, (n) is 7 (the total participants), and (r) is 3 (the positions being filled). Therefore, the correct calculation is:

[

7P3 = \frac{7!}{(7-3)!} = \frac{7!}{4!}

]

This shows that the first choice, having the expression (7!/(7-3)!), correctly represents the mathematical formulation