Final answer:
There are 665,280 different ways the top 6 racers can finish a race from a group of 12 racers, calculated using the permutation formula P(12,6) = 12! / (12-6)!.
Step-by-step explanation:
To determine how many different ways the top 6 racers in a video game finish if there are 12 available racers, we should use the concept of permutations. A permutation is an arrangement of objects in a specific order. Here, we consider the 'top 6' racers as our objects to arrange, and the '12 available racers' as the total number of objects we can choose from. We are looking for the number of ways to arrange 6 racers out of 12.
The formula for permutations when we are choosing r objects from a set of n objects and the order is important is expressed as P(n,r) = n! / (n-r)!. Applying this formula, we get:
P(12, 6) = 12! / (12-6)! = 12! / 6! = 479,001,600 / 720 = 665,280.
Therefore, there are 665,280 different ways for the top 6 racers to finish a race out of 12.