Final answer:
There are 6,840 different ways the students can come in the 1st, 2nd, and 3rd place in the race.
Step-by-step explanation:
When considering 20 students competing in a race and trying to determine the number of different ways they can come in 1st, 2nd, and 3rd place, we can use the concept of permutations. Permutations are used when the order of the arrangement matters. In this case, we can calculate the number of permutations using the formula:
P(n, r) = n! / (n - r)!
Where n is the total number of items (20 students) and r is the number of items we are selecting (3 places).
Plugging in the values:
P(20, 3) = 20! / (20 - 3)! = 20! / 17! = 20 * 19 * 18 = 6,840
So, there are 6,840 different ways the students can come in the 1st, 2nd, and 3rd place.