There are 55 participants in a regional dance competition. How many different ways can participants secure the gold, silver, and bronze positions?

This is a permutation problem. The number of ways to arrange n items taken r at a time is given by the permutation formula:

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

where:

• n is the total number of items
• r is the number of items to arrange
• "!" denotes factorial, which is the product of all positive integers up to that number

In this case, we have 55 participants and 3 positions (gold, silver, and bronze), so we want to find the number of ways to arrange 55 participants taken 3 at a time.

So, we calculate:

P(55, 3) = 55! / (55-3)! = 132,600

So, there are 132,600 different ways the participants can secure the gold, silver, and bronze positions.