Probability

Lesson 9

Math

Unit 8

10th Grade

Lesson 9 of 10

Objective


Describe and apply the counting principle and permutations to contextual and non-contextual situations.

Common Core Standards


Core Standards

  • S.CP.B.9 — Use permutations and combinations to compute probabilities of compound events and solve problems.

Criteria for Success


  1. Describe that a permutation is a sample space where the order in which the events occur matters.
  2. Calculate the total number of permutations when repetition is allowed by calculating $$ n^x$$, where $$x$$ is the number of positions you have available and $$n$$ is the number of choices you have for each of those positions. (This is a form of the counting principle.)
  3. Calculate the total number of permutations when repetition is not allowed by calculating $$n!$$, where $$n$$ is the number of choices you have for each position. $$ n! $$ Is represented algebraically as $$(n)(n-1)(n-2)(n-3)…$$ until the last binomial is equal to $$1$$.
  4. Calculate the permutations when choosing $$k$$ from a set of $$n$$ items—that is, given $$k$$ different positions available and $$n$$ set of items where $$ k<n$$, use the formula $$_nP_k=\frac{n!}{(n-k)!}$$.

Tips for Teachers


  • This lesson covers S-CP.9 (+) standard, which is a plus standard and therefore an optional lesson in this unit. 
  • It may seem counterintuitive to study permutations before combinations, but the formulas for calculation build from permutations (where order is important) to combinations (where order is not important). 
  • Anchor Problem #2 and Anchor Problem #3 build upon one another but are separated into two different problems because they serve different goals. 
  • EXTENSION: Use dy/dan, a blog by Dan Meyer, “My Lesson Plan: The Door Lock” to do a larger task than Anchor Problem #2 and Anchor Problem #3, where there is only a five-number combination lock.
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Anchor Problems

25-30 minutes


Problem 1

Using the digits 1, 2, and 3, write an expression to answer the following questions

  1. How many different three-digit numbers can you make when you can use each digit more than once?
  2. How many different three-digit numbers can you make when you can only use each digit once?

Guiding Questions

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Problem 2

Your phone requires you to use five numbers for the code, but you only can use numbers 1-5 for the code.

The first combination you come up with is:

1 2 3 4 5

You then realize how "insecure" this number might be since anyone could guess it. Figuring this out you wonder, "How many possibilities are there for combinations if I only use each number once?"

Guiding Questions

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Problem 3

You change the phone settings so that you only need a three-digit code to open the lock.

This means you have 5 numbers to choose from and 3 positions to put those numbers in, which is represented by $$_5P_3$$. How many different combinations can you get if you have three positions and a choice of five numbers with no repetition?

Guiding Questions

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Target Task

5-10 minutes


Problem 1

The combination for a lock consists of three numbers.

  1. If the numbers can be repeated, how many different combinations are there? Explain your answer.
  2. If the numbers cannot be repeated, how many different combinations are there? Explain your answer.

References

EngageNY Mathematics Precalculus and Advanced Topics > Module 5 > Topic A > Lesson 2Exit Ticket, Question #1

Precalculus and Advanced Topics > Module 5 > Topic A > Lesson 2 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Modified by Fishtank Learning, Inc.

Problem 2

Jacqui is putting together sets of greeting cards for a school fundraiser. There are four different card options, two different colored envelopes, and four different sticker designs. A greeting card set consists of one type of card, one color for the envelopes, and one sticker design. How many different ways can Jacqui arrange the greeting card sets? Explain how you determined your answer. 

References

EngageNY Mathematics Precalculus and Advanced Topics > Module 5 > Topic A > Lesson 2Exit Ticket, Question #2

Precalculus and Advanced Topics > Module 5 > Topic A > Lesson 2 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Additional Practice


The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

  • Include problems that ask for a probability of choosing the right random permutation, applying a “time per permutation,” and identifying the amount of time. (Using a movie clip where a device is cracking an alarm system code (how fast is the machine working) would be ideal.)

Next

Describe and apply the counting principle and combinations to contextual and non-contextual situations.

Lesson 10
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Lesson Map

A7CB09C2-D12F-4F55-80DB-37298FF0A765

Topic A: Conditional Probability and the Rules of Probability

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