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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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# 1. Two headed robots

What happens if the director changes their mind and asks for two headed robots?

## Video transcript

- Great job completing the last tutorial! You made both the director
and the producer very happy. Sometimes our directors change
their minds when they see their drawings translated
to 3D computer graphics. In fact, sometimes our directors
change their minds a lot, and we need to be able to
adapt to those changes. So let's say we have a
director that wants a robot with two heads, but each
head has to be different. That is, the director doesn't want a robot with two copies of the same head. Now a robot consists of two
heads, a body, and arms. We'd like to know how
many of those we can have if we have H heads, B bodies, and A arms. We saw that by using a Tree
Diagram we could multiply the number of each kind of part together to get the total number of combinations. Well, we have H choices
for the first head, but since the second
head has to be different, we only have H minus one
choices for the second head. As before, we have B choices for bodies and A choices for arms, for a total of H times H minus one times B times A combinations. So it's all about multiplying together all the choices you have at each stage. Mathematicians make up
words for everything. They use the word combinatorics to refer to this type of counting. The next exercise will give you a chance to practice with other combinations.