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- Good work! I left you with this question: Why are there exactly six combinations of each cast when you select three actors from a group of four? To get a feeling for what's up, let's look at the first two boxes. Notice that in the first cast, or box, we have all possible orderings of A, B, and C. Mathematicians call each of these orderings a permutation. Now, the number of permutations is represented by the number of rows in each box. The same is true in the second cast involving A, B, and D. How many orderings, or permutations, are there of three things? We saw earlier that there are three factorial, or six permutations. Bingo! That's it! To keep order for mattering, we need to take the total number of combinations where order does matter and divide that by all the possible permutations. So when choosing three actors from four actors total, we can write our calculation as four times three times two over three factorial, which equals four. Let's do another example and figure out how many casts we can form with three actors from a pool of six actors. In this case, there would be six times five times four. We then have to again divide by three factorial to keep the order from mattering, leaving six times five times four over three factorial, which equals 20. So there are 20 different casts in this case. Use the next exercise to get some practice with some other examples. And perhaps you'll recognize our good friend M-O in the cast. (robotic sounds)