There are 10 people at a meeting and there are 3 different door prizes to give out. How many ways can we select 3 people to receive the door prizes?

To determine the number of ways to select 3 people from 10 to receive different door prizes, we can think of it as arranging 3 specific items from a group of 10. This is a permutation problem because the order in which the prizes are given matters. The formula for permutations is given by P(n, r) = n! / (n - r)!, where n is the total number of items (people) and r is the number of items to arrange (prizes). Plugging in the values, we get P(10, 3) = 10! / (10 - 3)! = 10! / 7! = 10 × 9 × 8 = 720. Therefore, there are 720 ways to select and arrange 3 people to receive the different door prizes! 🎉 Now, let’s dive into some historical nuggets! Did you know that the concept of permutations and combinations can be traced back to the works of mathematicians in ancient India? They used these concepts in astronomy as well as in trading, showcasing the practical applications of these mathematical principles. Want to see how this applies in the real world? Think of award shows! When selecting winners from a group of nominees for various categories, the organizers use similar permutation and combination principles to ensure that there are unique recipients for each award. It's all about making every recognition special! 🏆