Hi there! Thank you for the opportunity to help you with your question!
In this problem, you have to choose one band as the favorite. Since there are five bands, there are five possibilities for the favorite. Now, once you have picked a band as your favorite, that only leaves four different bands as your possible second-favorite, there are four possibilities for your second place. Finally, by the same logic, there are only three bands left, or three possibilities for your third place.
The total number of possibilities is actually the product of all these, 5 x 4 x 3 = 60, because of all the mixing and matching.
Another (way longer) way of solving this problem is to try to number all possibilities. Assume the name of the bands are A, B, C, D, E. Then all possibilities for 1st, 2nd and 3rd place can be written as (with the first letter being favorite, 2nd letter second place, 3rd letter - third place)