Question

In a race with 7 runners, how many different orderings of 1st, 2nd, and 3rd place can there be?

1) 42
2) 210
3) 5040
4) 720

Answer 1

There can be 7ₚ₃ = 210 different orderings of 1st, 2nd, and 3rd place in a race with 7 runners.

Step-by-step explanation:

To find the number of different orderings for the top three positions in a race with 7 runners, permutations are employed. Permutations determine the arrangements of a specific number of items from a larger set. In this case, we're interested in arranging the top three positions out of 7 runners.

The formula for permutations is expressed as nₚᵣ = n! / (n - r)!, where n represents the total number of items and r denotes the number of items being arranged.

Applying the permutation formula for selecting 3 runners out of 7: 7ₚ₃ = 7! / (7 - 3)! = 7! / 4! = (7 × 6 × 5) / (4 × 3 × 2 × 1) = 210. Therefore, there are 210 different ways to arrange the 1st, 2nd, and 3rd positions among the 7 runners in the race.