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## Pixar in a Box

### Unit 13: Lesson 2

Counting crowds- Start here!
- 1. Two headed robots
- Counting two-headed robots
- 2. Snake bots
- Building snake bots
- 3. Calculating factorials
- Calculating factorials
- 4. Casting problem
- Counting casts 1
- 5. Does order matter?
- Counting casts 2
- 6. Binomial coefficient
- Combinations

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# 5. Does order matter?

Why do we divide the number of combinations by the number of permutations?

## Video transcript

- 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)