Objective
Describe and apply the counting principle and combinations to contextual and non-contextual situations.
Common Core Standards
Core Standards
The core standards covered in this lesson
S.CP.B.9— Use permutations and combinations to compute probabilities of compound events and solve problems.
High School — Statistics and Probability
S.CP.B.9— Use permutations and combinations to compute probabilities of compound events and solve problems.
Criteria for Success
The essential concepts students need to demonstrate or understand to achieve the lesson objective
- Describe that a combination is a sample space where the order in which the events occur does not matter.
- Calculate the total number of combinations by multiplying the number of items in one category by the number of items in each of the other categories. This is the fundamental counting principle.
- Describe how the process and formulas for permutations and combinations are similar and different.
- Calculate the combinations 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 $$_nC_k= \frac{n!}{k!(n-k)!}$$.
Tips for Teachers
Suggestions for teachers to help them teach this lesson
This lesson covers S-CP.9 (+) standard, which is a plus standard and therefore an optional lesson in this unit.
Fishtank Plus
Unlock features to optimize your prep time, plan engaging lessons, and monitor student progress.
Anchor Problems
Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding
25-30 minutes
Problem 1
You bring your niece to Tasty Burger for lunch over break. She is notorious for not making a decision quickly, so you want to know just how many possibilities there are for putting together a Kids Meal. Here is the menu:
Kid's Meal
- Junior Hamburger, Junior Cheeseburger, Nuggets, Hot Dog, or Grilled Cheese
- Fries
- Apple Juice, Fruit Punch, 1% milk, or Chocolate milk
$5.95
How many choices does your niece have for a complete meal from Tasty Burger?
Guiding Questions
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Problem 2
Your friend has a set of three cards with the letters $$A$$, $$B$$, and $$C$$ on them.
Situation 1: He says, “Choose one card and then another card. Order of the letters is important.”
Situation 2: He tells you, “Choose two cards. Order of the letters is not important.”
- Which situation represents a permutation? A combination? Explain your reasoning.
- By what factor are the numbers of outcomes different in each of these situations?
Guiding Questions
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Problem 3
Seven speed skaters are competing in an Olympic race. The top three skaters will move on to the semi-finals. It doesn’t matter whether you are first, second, or third to advance to the next round. How many different “top three” groups can be selected?
Guiding Questions
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References
EngageNY Mathematics Precalculus and Advanced Topics > Module 5 > Topic A > Lesson 3—Example 1
Precalculus and Advanced Topics > Module 5 > Topic A > Lesson 3 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 USlicense.Accessed Dec. 2, 2016, 5:15 p.m..
Modified by Fishtank Learning, Inc.
Target Task
A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved
5-10 minutes
Problem 1
Timika is a counselor at a summer camp for young children. She wants to take 20 campers on a hike and wants to choose a pair of students to lead the way. In how many ways can Timika choose this pair of children?
References
EngageNY Mathematics Precalculus and Advanced Topics > Module 5 > Topic A > Lesson 3—Exit Ticket, Question #1
Precalculus and Advanced Topics > Module 5 > Topic A > Lesson 3 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 USlicense.Accessed Dec. 2, 2016, 5:15 p.m..
Problem 2
Sean has 56 songs on his MP3 player. He wants to randomly select 6 of the songs to use in a school project.
- How many different groups of 6 songs could Sean select?
- Did you calculate the number of permutations or the number of combinations to get your answer? Why did you make this choice?
References
EngageNY Mathematics Precalculus and Advanced Topics > Module 5 > Topic A > Lesson 3—Exit Ticket, Question #2
Precalculus and Advanced Topics > Module 5 > Topic A > Lesson 3 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 USlicense.Accessed Dec. 2, 2016, 5:15 p.m..
Problem 3
A fast-food restaurant has the following options for toppings on their hamburgers: mustard, ketchup, mayo, onions, pickles, lettuce, and tomato. In how many different ways could a customer choose four different toppings from these options?
References
EngageNY Mathematics Precalculus and Advanced Topics > Module 5 > Topic A > Lesson 3—Exit Ticket, Question #3
Precalculus and Advanced Topics > Module 5 > Topic A > Lesson 3 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 USlicense.Accessed Dec. 2, 2016, 5:15 p.m..
Problem 4
Seven colored balls (red, blue, yellow, black, brown, white, and orange) are in a bag. A sample of three balls is selected without replacement. How many different samples are possible?
References
EngageNY Mathematics Precalculus and Advanced Topics > Module 5 > Topic A > Lesson 3—Exit Ticket, Question #4
Precalculus and Advanced Topics > Module 5 > Topic A > Lesson 3 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 USlicense.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.
- Open Middle Combinations
- Reason and Wonder Coin Combinations
- Illustrative Mathematics Random Walk III
- EngageNY Mathematics Precalculus and Advanced Topics > Module 5 > Topic A > Lesson 3
- EngageNY Mathematics Precalculus and Advanced Topics > Module 5 > Topic A > Lesson 4—(Focuses on the difference between permutations and combinations)
- Open Middle Combinations and Permutations
- Illustrative Mathematics Alex, Mel, and Chelsea Play a Game
- The Secondary Classroom Can Be Fun Too Lies in Advertising - Analyzing a Real World Problem through Permutation/Combinations
Lesson 9