A gymnastics competition has 9 gymnasts. How many different top-4 rankings are possible?

To solve this problem, we use the concept of permutations in combinatorics. A permutation is an arrangement of objects in a specific order. The number of permutations of n objects taken r at a time is given by the formula:

P(n, r) = n! / (n-r)!

where "!" denotes factorial, which is the product of all positive integers up to that number.

In this case, we have 9 gymnasts (n=9) and we want to find out the number of ways we can rank the top 4 (r=4).

So, we plug these values into our formula:

P(9, 4) = 9! / (9-4)!

= 9! / 5!

= (987654321) / (54321)

= 987*6

= 3024

So, there are 3024 different top-4 rankings possible in this gymnastics competition.