Step 1 :This problem is about permutations. We have 12 singers and we want to know how many ways we can arrange 5 of them. The order matters here because the prizes are different for each position from first to fifth.
Step 2 :We can use the formula for permutations to solve this problem. The formula is \(P(n, r) = \frac{n!}{(n-r)!}\), where \(n\) is the total number of items, and \(r\) is the number of items to choose.
Step 3 :Substitute \(n = 12\) and \(r = 5\) into the formula, we get \(P(12, 5) = \frac{12!}{(12-5)!}\).
Step 4 :Calculate the factorial of 12 and 7, we get \(12! = 479001600\) and \(7! = 5040\).
Step 5 :Divide 479001600 by 5040, we get 95040.
Step 6 :So, there are \(\boxed{95040}\) ways the singers can finish first through fifth.