Hello Everyone, There Are 128 Runners From 27 Countries

Question

Post 1hello Everyonethere Are 128 Runners From 27 Countries Competing There are 128 runners from 27 countries competing in a track meet. The betting pool needs to know how many outcomes there may be for the top 3 finishers. To determine the number of possible outcomes for the top 3 positions, we use permutations because the order matters — the first, second, and third places are distinct. The formula for permutations is P(n, r) = n! / (n - r)! where n is the total number of options, and r is the number of selections. Applying this to our scenario, for the top 3 finishers among 128 runners, the calculation is: P(128, 3) = 128! / (128 - 3)! = 128 × 127 × 126. This product gives us the total number of different outcomes for the top 3 positions. Specifically, 128 × 127 × 126 equals 2,046,456. Therefore, there are 2,046,456 possible different outcomes for the top three finishers among the 128 runners. Understanding Permutations in Context Permutations are critical in scenarios where the order of selection matters, such as determining the rankings in a race. In this case, since each position (first, second, third) is distinct, permutations accurately model the problem. If the order did not matter, such as in selecting a group without regard to position, combinations would be appropriate. Additional Considerations While permutations provide the answer for the top three outcomes, calculating these factorials manually is impractical for very large numbers like 128!. Instead, calculators or software designed for combinatorial calculations are used. These tools can handle large factorials efficiently and prevent computational errors associated with manual calculations. Post 2 Hello everyone, During a selection for a special duty, the judges had to pick 5 people from 15 contestants In this scenario, the problem involves choosing a group of 5 contestants from a pool of 15 applicants. Since the order of selection does not matter—selecting {Alice, Bob, Charlie, David, Eva} is the

Dr. Jack HW Helper · Accepted Answer

Post 1hello Everyonethere Are 128 Runners From 27 Countries Competing There are 128 runners from 27 countries competing in a track meet. The betting pool needs to know how many outcomes there may be for the top 3 finishers. To determine the number of possible outcomes for the top 3 positions, we use permutations because the order matters — the first, second, and third places are distinct. The formula for permutations is P(n, r) = n! / (n - r)! where n is the total number of options, and r is the number of selections. Applying this to our scenario, for the top 3 finishers among 128 runners, the calculation is: P(128, 3) = 128! / (128 - 3)! = 128 × 127 × 126. This product gives us the total number of different outcomes for the top 3 positions. Specifically, 128 × 127 × 126 equals 2,046,456. Therefore, there are 2,046,456 possible different outcomes for the top three finishers among the 128 runners. Understanding Permutations in Context Permutations are critical in scenarios where the order of selection matters, such as determining the rankings in a race. In this case, since each position (first, second, third) is distinct, permutations accurately model the problem. If the order did not matter, such as in selecting a group without regard to position, combinations would be appropriate. Additional Considerations While permutations provide the answer for the top three outcomes, calculating these factorials manually is impractical for very large numbers like 128!. Instead, calculators or software designed for combinatorial calculations are used. These tools can handle large factorials efficiently and prevent computational errors associated with manual calculations. Post 2 Hello everyone, During a selection for a special duty, the judges had to pick 5 people from 15 contestants In this scenario, the problem involves choosing a group of 5 contestants from a pool of 15 applicants. Since the order of selection does not matter—selecting {Alice, Bob, Charlie, David, Eva} is the

Hello Everyone, There Are 128 Runners From 27 Countries

Post 1hello Everyonethere Are 128 Runners From 27 Countries Competing

Understanding Permutations in Context

Additional Considerations

Post 2 Hello everyone, During a selection for a special duty, the judges had to pick 5 people from 15 contestants

Applications of Combinations

Significance in Real-world Contexts

References