In a marathon race with 50 runners, you want to predict exactly which runner will finish first, which runner will finish second, and which runner will finish third. In how many different ways can the top three positions occur?

Introduction / Context:
This problem is about ordered arrangements, which are handled using permutations. We are not just choosing which three runners are in the top three, but also specifying the exact finishing order: first, second, and third. This makes the problem more than a simple combination. Understanding when to switch from combinations to permutations is crucial in many real life ranking and arrangement problems.

Given Data / Assumptions:

• Total number of runners in the marathon: 50.
• We are concerned with the first three finishing positions: first, second, and third.
• All 50 runners are distinct individuals.
• The order of finish matters, since first place is different from second or third.
• No ties are considered; exactly one runner holds each of the top three positions.

Concept / Approach:
Because we care about both who is in the top three and the exact order, we use permutations rather than combinations. The number of ways to choose and arrange r items from n distinct items is given by:
nPr = n * (n - 1) * (n - 2) * ... up to r factors. Here, n = 50 and r = 3. So we must compute 50P3, which counts all ordered triples of distinct runners.

Step-by-Step Solution:
Step 1: There are 50 choices for who comes first. Step 2: After choosing the first place runner, 49 runners remain, so there are 49 choices for second place. Step 3: After that, 48 runners remain, giving 48 choices for third place. Step 4: Apply the multiplication principle: total ways = 50 * 49 * 48. Step 5: Compute 50 * 49 = 2450. Step 6: Then multiply 2450 * 48 = 117600. Step 7: Therefore, there are 117600 possible ordered outcomes for the top three positions.

Verification / Alternative check:
We can express the same reasoning using the permutation notation 50P3. The formula for permutations is 50P3 = 50! / (50 - 3)!. That gives 50! / 47!, which simplifies to 50 * 49 * 48, the same as our stepwise counting. A calculator that supports nPr will also confirm that 50P3 = 117600, validating the result.

Why Other Options Are Wrong:
19600: This is equal to 50C3 and would be correct only if order did not matter. 125000: Looks like 50^3, which would allow repetition, not valid in a race with unique finishers. 60000: A random large number without a correct combinatorial interpretation for this scenario. Only 117600 correctly counts ordered distinct top three outcomes from 50 runners.

Common Pitfalls:
A very common mistake is to use combinations, 50C3, instead of permutations. That counts only which three runners appear in the top three, ignoring their finish order. Another error is to miscalculate 50 * 49 * 48 due to arithmetic slips. Breaking the multiplication into two steps and checking results with a calculator helps avoid such mistakes. Also, including repetitions such as allowing the same runner to appear multiple times in the top three is logically wrong and inflates the count incorrectly.

Final Answer:
The number of different ways in which the top three positions can be filled by 50 runners is 117600.