Final answer:
There are 30 different ways for 6 athletes to finish first or second in a track meet, calculated by the permutation formula P(6, 2) = 6! / (6-2)! = 30. The coach's new stopwatch is helpful due to its precision in timing close sprint finishes.
Step-by-step explanation:
To determine how many different ways 6 athletes at a track meet can finish first or second, we need to calculate the number of permutations for the first two places out of the 6 athletes. Since order matters in ranking first and second, this is a permutation problem.
The formula for permutations is P(n, k) = n! / (n-k)!, where n is the total number of items to choose from, k is the number of items to choose, and ! represents factorial. Here, n=6 athletes and k=2 places. Thus, the permutations for the first and second place are P(6, 2) = 6! / (6-2)! = 6x5 = 30 different ways.
The coach's new stopwatch with an uncertainty of ±0.05s will be helpful in timing the sprint team, especially given that the difference in times between the first and second place sprinters at the school's last track meet was just 0.03 seconds. Such a precise timing device will allow for more accurate measuring of the sprinters' performance.