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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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# Start here!

Overview of this topic

# Ready to dive into some more math?

In the first lesson you learned that it was possible to build 1000

*possible*robots using only a handful of parts. Now suppose the director only asks for a cast of 6*different*robots from the set of 1000 possible robots. How many possible casts would this result in?This question is easy IF you know how to think about it. In this lesson we are going to develop a really powerful formula we can use to answer questions like this. It's known as the

**binomial coefficient**:Specifically we'll want to answer this question: given n possible robots how many different casts could we make of size k?

To get there we are first going to introduce

**permutations**by counting the number of*different*robotic snakes we can build by rearranging the*same set*of parts.Finally we'll combine the ideas of permutations and combinations to arrive at the general form of the binomial coefficient:

# What do I need to know before starting?

- You should have finished the first lesson
- You should be comfortable with algebra basics
- Remember, you can always work through part of the material

# Grade level standards

Below are the relevant Common Core State Standards for both lessons in this topic:

## Lesson 1: Building crowds

Appropriate for all ages and introduces the counting principle (Grade 4-7 appropriate).

#### Grade 6

- CCSS.MATH.CONTENT.6.EE.A.2 Relevant because we are evaluating expressions in which letters stand for numbers.

#### Grade 7

- CCSS.7.SP.C.8.B
- Relevant because we explain compound events using tree diagrams (counting principle).

## Lesson 2: Counting Crowds

This lesson begins with permutations and reaches the high school level. (Grade 7+ appropriate)

#### High School

- HSS-CP.B.9
- Relevant because we are using combinations to solve problems involving compound events.

- HSA-APR.C.5
- Relevant because binomial coefficients are terms in the binomial theorem expansion.

## Want to join the conversation?

- Do we have to like math to do this?(11 votes)
- No, but hopefully by the end of it you will.(21 votes)

- What if you're not in high school, and you're not in middle school yet? Can you still tackle these lessons?(3 votes)
- You can if you believe that you can and you are willing. Those were just grade recommendations.(6 votes)

- What is this about... ?

Sorry but Ican't undeerstand..(4 votes) - What if the math is a little (a lot) harder for a 4th grader? will this make me smarter? Write soon!(1 vote)
- Working hard to understand it is far more important than "being smart" so you can understand it easily. Most of the problems that people care about solving have not been solved yet, so the more practice you get doing hard things, the better off you will be.(4 votes)

- the Working hard to understand it is far more important than "being smart" so you can understand it easily. Most of the problems that people care about solving have not been solved yet, so the more practice you get doing hard things, the better off you will be.(1 vote)
- it looks like one in 2,3,4,5,6,7,8,9or 10... Desn't it ?(0 votes)
- Is this about factorials, permutation formula, and combination formula? Will it help me with those things?(0 votes)