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Permutations and Combinations
Resource ID#: 71610
Primary Type: Lesson Plan
This document was generated on CPALMS - www.cpalms.org
This is a seventh grade lesson that should follow a lesson on simple probability. This is a great
introduction to compound probability and a fun, hands-on activity that allows students to explore the
differences between permutations and combinations. This activity leads into students identifying
situations involving combinations and permutations in a real-world context.
Subject(s): Mathematics
Grade Level(s): 7
Intended Audience: Educators
Suggested Technology: Scientific Calculator
Instructional Time: 2 Hour(s)
Freely Available: Yes
Keywords: permutations, combinations, compound events
Instructional Component Type(s): Lesson Plan
Resource Collection: FCR-STEMLearn Algebra
ATTACHMENTS
Permutation Combination Activity Recording Sheet.docx
Permutations and Combinations guided practice.docx
Permutations and Combinations guided practice answer key.docx
Permutation and Combinations Activity Cards.docx
Permutation Combination Homework.docx
Permutation Combination Homework Key.docx
Permutations and Combinations Exit Quiz.docx
Permutation Combination Exit Quiz Key.docx
LESSON CONTENT

Lesson Plan Template:
General Lesson Plan

Learning Objectives: What should students know and be able to do as a result of this
lesson?
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Students will be able to distinguish the similarities and differences between permutations and
combinations.
Students will be able to correctly choose when to use permutations and combinations in order to
determine the probability of compound events and to solve problems.
Students will be able to calculate permutations and combinations to then find the probability of a
compound event or to solve a problem.
Prior Knowledge: What prior knowledge should students have for this lesson?
MAFS.7.SP.3.8a - Understand that, just as with simple events, the probability of a compound event is the
fraction of outcomes in the sample space for which the compound event occurs.
MAFS.7.SP.3.7 - Develop a probability model and use it to find probabilities of events. Compare probabilities
from a model to observed frequencies; if the agreement is not good, explain possible sources of the
discrepancy.

Guiding Questions: What are the guiding questions for this lesson?
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What is a compound event?
How does a compound event compare to an independent event?
What is a permutation?
What do we do with combinations and permutations?
How can we use combinations and permutations to solve real world problems?
Teaching Phase: How will the teacher present the concept or skill to students?
Day 1:
The lesson will begin with the student activity Permutation and Combinations Activity Cards.docx. Have
students answer the questions on the activity recording sheet Permutation Combination Activity Recording
Sheet.docx. When the activity is complete, the teacher will facilitate a discussion about the results of the
activity with the emphasis on deciding when order matters (permutations) and when order doesn't matter
(combinations). Use the information provided below for each activity to facilitate the whole class discussion
and assist students in seeing the differences between permutations and combinations.
For the activity, students will be placed in groups of three, four or five depending on the number of students in
your class. You can choose to group students based on proximity of one another for ease or based on student
ability. To help effectively manage the 4 activities, separate the activities onto separate card stock and have all
groups complete each part at the same time (e.g. each group does activity one at the same time and then moves
through the other activities as they complete each one). This allows the activities to proceed one at a time for
the teacher’s pace.
To teach the lesson as designed, all 4 activities require a fairly large flat surface to remain visible for the
discussion. The discussion is the most important part of the lesson as students explain and think through the
patterns. Since the teacher is the only one who knows their classroom setup, please direct students to place the
activities in rows or corners, each group will complete a Permutation Combination Activity Group Sheet to
record their findings.
Activity 1 should take less than 5 minutes. No two pairs of students will have the same color combination. The
order does not matter because it is simply a pair (group) - there is no need for students to greet each other a
second time. The teacher will know when the activity is done correctly when there is [(number of students in
group) – 1] of each color. The illustration below reflects a group of 5 people:
Activity 2 should take between 5 - 10 minutes. First, the students (as a group) select any three colors. Students
will make groups using all 3 colors in different orders. When students expand their groupings to allow all
patterns (order matters) the groupings will yield six permutations.
Activity 3 should take 10-15 minutes. Each group chooses all five colors and works together to create unique
sets of three colors. No set can have the same colors combined with one another, creating combinations. (It is
much more difficult to complete Activity 3 than
Activity 4, because finding the duplicates can be difficult.)
See below:
Activity 4 should take 10-15 minutes. In this activity, each group chooses four different colors and work
together to create all the sets possible using three colors at a time. The colors in the set do not have to be unique
but the positions of the colors must be different. It might be easier to find the 24 permutations using a
systematic list.
Some students will be able to form combinations quicker than permutations and vice-versa. The only help
teachers should provide is encouraging students to develop a pattern for Activity 3 and Activity 4 and to check
if their totals are correct. The teacher should encourage students to look for patterns within the pattern. The
teacher can quickly tell if the sets are correct by counting the individual colors in each activity – they should all
be equal. If they are struggling with a pattern, encourage them to begin and then work in a methodical manner –
there should be a 'method to their madness.'
Suggest they start with one color first, then build combinations that start with that color. Use color 1 and color
2, then all the combinations that can be made. Then color 1 and color 3; color 1 and color 4; and color 1 and
color 5. Then they start with color 2 first, and so on.
Patterns appear when we begin to look for them. Let the students think it through and they will understand
instead of simply showing them how to plug numbers into a formula.

Guided Practice: What activities or exercises will the students complete with teacher
guidance?
Day 2:
The guided practice/homework sheet, Permutation Combination Guided Practice, is designed to be duplexed
and cut in half.
Once the activity is complete and the Permutation Combination Activity Group Sheet is collected the
teacher will facilitate a discussion among the groups. Let the students talk about the activity, the outcome and
the process. What did they learn? What patterns did they see and how did they discover them? Have them walk
around the tables and look at the other teams' work. No two tables will look exactly alike.
The teacher will introduce and explain factorials. Work out a couple of quick scenarios. 5! = 5 X 4 X 3 X 2 X 1
= 120 etc. Next relate factorials to the counting principle - they both multiply everything together to get the
total possible outcomes. But with factorials you know what the numbers are based on a pattern. Make sure to
note that 0! = 1 since we can never divide by 0 and permutations where n = k always have zero factorial in the
denominator.
If a factorial provides the answer for total outcomes, can we use that knowledge to determine how many
outcomes there are when we only want to use some of the outcomes? If factorial is everything and we don't use
everything, we need to exclude what we don't use. Since we formed factorials with multiplication we will use
division to exclude the part we don't want. In simple terms, permutations are the total of all possible
outcomes divided by the total of the outcomes not selected. We use factorials to find those totals.
Have the students compare the even numbered activities with the odd numbered activities. When they discuss
the difference between groups, introduce the idea that order matters. When all arrangements are represented
because order matters it is called a permutation. Show the formula
scenarios from the activity that are permutations (Activity 2 and 4).
, then work out the
Note for students that P(n,k) is the same as
Now introduce Combinations. Combinations are exactly like Permutations except we have to remove the nonunique sets because order does not matter, we are simply forming groups. So how much do we remove? The
space of the duplicates, the number of element in each set Factorial. Introduce the formula
, then work out the scenarios from the activity that are combinations (Activity 1
and 3). Again, note for students that C(n,k) is the same as
Discuss the definition for the word combination in general. Most students define combinations as all the ways
different elements can be combined but that is actually permutations! Remind them that Permutations >
Combinations – but wait a minute – is it always? Make sure students know that When n and k = 1,
Permutations actually = Combinations.
What about using n or k = zero – is that possible? If there is nothing to choose from, can anything be chosen?
Similarly, if something exists can groups of nothing be created? Work out any permutation problem where k =
n – 1 and k = n. Both yield the same number of outcomes. Why? It is easy to see that we are multiplying the
same numbers and dividing by 1 but where does the 1 come from? In one instance, the 1 comes from the
difference and 1! = 1 and in the other instance, the difference is 0 and 0! =1. So are they the same? If not, how
are they different? A very quick visual is to redo Activity 2 to use the same 3 colors but create groups of 2.
Students should recognize that the number of outcomes are the same but both the arrangement of the pieces and
the number of pieces used is different.
When beginning calculations, it is good to use factorials in expanded form for the students who may be
struggling a bit and mark the numbers off before moving on to using factorials and crossing the entire factorial
off.
Once this class discussion is complete, allow students to work in pairs to complete the Guided Practice
worksheet: Permutations and Combinations guided practice answer key.docx Permutations and Combinations
guided practice.docx
As students are working on the guided practice, the teacher should move around the room and assist students as
they need help. Once all groups are finished, review the solutions and discuss any questions that students have
or clear up any misconceptions that you witnessed when moving around the room.

Independent Practice: What activities or exercises will students complete to reinforce the
concepts and skills developed in the lesson?
The students will complete the attached homework assignment, independently. Allow students to begin
working on independent practice with the remaining time left in the class period. If students do not finish,
instruct them to finish for homework.Permutation Combination Homework.docx Permutation Combination
Homework Key.docx
While students are working independently, the teacher should be moving around the room checking on student
understanding and answering any questions.

Closure: How will the teacher assist students in organizing the knowledge gained in the
lesson?
The teacher will bring the whole class back together for a review of the lesson to reinforce what has been
learned.
The teacher should review the answers of the guided practice and answer any lingering student questions.

Summative Assessment
The teacher will determine if students have reached the learning target by checking the self-assessment section
of the guided practice and having students complete an exit quiz. Permutation Combination Exit Quiz Key.docx
Permutations and Combinations Exit Quiz.docx

Formative Assessment
During the Activity, the teacher can monitor student progress by walking around the room and observing
students. Try to offer encouragement rather than answers and let students work through each part as a group.
After the Activity, the teacher will be able to check for understanding through whole class discussion, in the
self assessment section of the guided practice, and by checking the exit quiz.
Scale:
The Self-Assessment section of the guided practice is where each student rates their knowledge of the concepts
listed in the learning goal from the beginning of the lesson until the guided practice is complete. This lesson
will use a generic scale. Post a permanent display in the classroom in a prominent spot. The only item the
teacher will need to change is the daily learning goal. If unit goals are needed, for this lesson simply add a unit
goal about Compound Probability.
Since this is the first time factorials, combinations and permutations have been taught:
1 for Beginning, "I have no prior knowledge of the learning goal, I need help." would be appropriate.
Just before the student turns the guided practice over assess student understanding. Most students should now
be able to enter a 2 for Developing, "I can solve simple problems with the learning goal, I may need help” or a
3 for Applying, "I can solve complex problems with the learning goal, I don’t need help.”
Remind them to assess one last time before the guided practice is collected. If they are struggling, encourage
them to fill in the part that says I don’t understand …. At the end of the lesson, many students will enter a 4 for
Expert, "I can solve complex problems with the learning goal, I can teach another.”

Feedback to Students
Students will receive feedback as a group during the activity and discussion, then individually through guided
practice and homework
ACCOMMODATIONS & RECOMMENDATIONS

Accommodations:
The teacher may choose to place struggling students in pre-determined groups so that they are grouped with a
high ability student and at least one average ability student.
The teacher should be conscientious of students with color-blindness, as these activities require students to
differentiate by color. A possible accommodation might be to label each colored slip with a distinct letter
designation to remove the need to see color (e.g. the blue paper have a B, the red paper an R).
Students with auditory processing issues may be provided with hard copies of any information the teacher
presents.

Extensions:
Students can create their own problem situations where permutations and combinations are appropriate tools.
Students can create their own and then share with a partner. Each person can then complete the partners
problems and compare solutions and rationale.


Suggested Technology: Scientific Calculator
Special Materials Needed:
brightly colored card stock in 5 colors pre-cut (see Further Recommendations for specific dimensions)
glue stick
timer

Further Recommendations:
Calculator Note:
Save the Scientific Calculators until a later date. This lesson focuses on students learning how to solve
combinations and permutations with paper and pencil providing a concrete foundation of the concepts.
Later, once they have a firm understanding, they can be shown the calculator functions.
Students get excited when solving combinations and permutations – it feels good to cross those numbers off!
Combinations and permutations are easy to set up.
When using a calculator, students are not able to see the patterns.
Materials:
Cut each sheet lengthwise in 1/2 inch strips and then cut those strips into 1 inch pieces. Use SMALL binder
clips to cut several sheets of cardstock at the same time. Put three clips on the left side of the sheets, oriented
lengthwise. Cut ½ inch strips. Take several of the ½ inch strips and put a binder clip on the end and then cut 1
inch pieces. There will be 187 pieces per sheet so 3 pieces of cardstock per color is the minimum quantity
needed for 6 groups. Cut out the activity cards and glue to 1/6 of a sheet of cardstock. Use a different color for
each of the four activities. Perfection is not required but try to make the pieces as uniform as possible.
Recommendation:
Select card stock colors that vary in hue. For some students, it is difficult to distinguish between yellow, orange
and green.
SOURCE AND ACCESS INFORMATION
Contributed by: Laura Quinton
Name of Author/Source: Laura Quinton
District/Organization of Contributor(s): Orange
Is this Resource freely Available? Yes
License: CPALMS License - no distribution - non commercial
Related Standards
Name
MAFS.7.SP.3.8:
Description
Find probabilities of compound events using organized lists, tables,
tree diagrams, and simulation.
a. Understand that, just as with simple events, the probability
of a compound event is the fraction of outcomes in the
sample space for which the compound event occurs.
b. Represent sample spaces for compound events using
methods such as organized lists, tables and tree diagrams.
For an event described in everyday language (e.g., “rolling
double sixes”), identify the outcomes in the sample space
which compose the event.
c. Design and use a simulation to generate frequencies for
compound events. For example, use random digits as a
simulation tool to approximate the answer to the question:
If 40% of donors have type A blood, what is the probability
that it will take at least 4 donors to find one with type A
blood?