Final answer:
The number of ways to arrange finalists in a singing competition for the first through fifth places is a permutation problem. Assuming there are 'n' finalists, the calculation is n × (n-1) × (n-2) × (n-3) × (n-4). This calculation reflects the counting principle without replacement.
Step-by-step explanation:
The problem described is an example of a permutation, where we want to find the number of ways to arrange a certain number of finalists in a singing competition for the top 5 positions. This is a classical permutations problem in combinatorics, a fundamental topic in probability and statistics.
Let's assume there are n finalists in total. The number of ways the top five can be chosen is given by the formula for permutations, which is P(n,5) = n! / (n-5)!. For the top position, we have n possibilities, for the second n-1, and so on, until we have n-4 possibilities for the fifth spot. Multiplying these together gives us the total number of ways the singers can finish in first through fifth place:
n × (n-1) × (n-2) × (n-3) × (n-4).
For example, if there were 10 finalists, there would be 10 × 9 × 8 × 7 × 6 = 30,240 different ways for the singers to finish in the top five spots. This calculation is based on the principle of counting without replacement since once a singer has been awarded a top spot, they cannot be selected for another spot.