Final answer:
There are 12,144 ways the 24 students in a spelling bee can win first, second, or third place, calculated using permutations formula 24P3, which equals 24 × 23 × 22.
Step-by-step explanation:
To determine how many ways 24 students can win first, second, or third place in a spelling bee, we need to calculate the number of permutations since the order in which the students place is important. The number of permutations of n distinct objects taken r at a time is calculated by the formula nPr = n! / (n-r)!. In this case, we have 24 students and we are looking to find the top 3, so r is equal to 3.
The calculation is then 24P3 = 24! / (24-3)!. Simplify the factorial notation by canceling out the common terms: 24! / 21! = 24 × 23 × 22. Thus, the number of ways the students can win first, second, or third place is 24 × 23 × 22 = 12,144 ways.
This type of problem is common in combinatorics, a branch of mathematics that deals with the counting, arrangement, and combination of objects. Understanding how to calculate permutations is important for solving such problems.