Permutations and Combinations - Introduction
What you will learn in this lesson on Permutations and Combinations?
Tell me in how many ways you (A) and your brother (B) can sit in two chairs before the TV?
2 ways, right?
AB and BA.
Now, you (a), your dad (b) and your mom (c) can sit in three chairs in how many ways?
Hmm, interesting but thoughtful!
Here it is:
abc, acb, bac, bca, cab, cba.
First, you occupy the first chair. Ask your dad and mom to exchange the other two seats in two ways.
Now, its your dad’s turn in the first chair. Now, you and mom will exchange the other two chairs in two ways.
Last, your mom will lead. Like to exchange seats with your dad in two ways? Try it anyway!
So, how many?
Six, in all.
Made it? Yes, you did!
Now, face this. More stuff to dig your brains deep
What if your teeth-grinder brother (d) joins in?
Hmm, big trouble? But, no escape!
You must let in your parents’ other half.
See how many ways you, your brother, your dad and your mom can sit in four chairs?
Here goes the list of
How many totally? My god! 24. That’s too many. Right? Ok.
Now I won’t ask you to add your friend Joe too.
Each of the above 24 sequences is called a permutation.
Permutations are also called arrangements.
We say 4 different persons denoted by letters a, b, c and d can be arranged in 24 ways.
We use a formula to find this. Including the formula, we learn other numerous cases in which the formula will be applied to produce various other formulas
If you wish to set off with your lesson on Permutations and Combinations, then click on the link below:
Permutations and Combinations
Or, if you wish to capture a terse overview of each Permutations and Combinations Formula, then go through each of the following header-links. You can also click the header-links to take you to the page on the specific Permutations and Combinations formula:
Fundamental Rule of Counting:
If there are m ways of doing one task, and n ways of doing another task, then the two tasks can be done one after other in m × n ways.
Factorial of n,n!
the number of ways in which n different things can be arranged by taking all at a time, when each thing can appear only once in every arrangement is called Factorial of n, denoted as n!
Each of the arrangements represented by n! is called a permutation.
Again each of the npr arrangements is also called a Permutation.
the number of ways in which n different things can be arranged (permuted) by taking r at a time, when each thing is allowed to repeat any number of times (upto a maximum number of r times) is nr
Each of the arrangements is a permutation
The number of ways in which n different things can be arranged by taking all at a time, if p things are same of one type, q things are same of a second type and the remaining of the n things are all different from each other is
Note that every arrangement is a permutation.
Circular PermutationsThe number of ways in which n different things can be arranged around a circle is :
(n – 1)!
Each of the arrangements around the circle is called a circular permutation.
Circular Permutations: Clockwise and Counter-clock wise
If the clock wise or anti-clock wise direction of permutations around a circle are not relevant, then the number of permutations around a circle is only
[(n – 1)!]/2.
Combinations:Suppose you (A) and your brother (B) wish to team up to play badminton against a group of two of your friends.
You and your bro can make how many teams?
Of course, only one team.
The team can be named as AB.
Will the team BA be different from the team AB?
You guessed it. It’s not!
Teams AB and BA are same, i.e., not different from each other. They are not different because the two members A and B, i.e. you and your brother are the same team members irrespective of whether the team is called AB or BA.
So, there we have a clear distinction between an arrangement and a group.
As a permutation AB and BA are different from each other, but as a group (combination) AB and BA are same.
From n different things, by taking any r things at a time, the number of groups that can be formed is ncr
A useful alternative definition of ncr
The number of ways in which any r things can be selected out of n different things is ncr
Important formulas on Combinations:
- nc0 = 1
- ncn = 1
- ncr = ncn-r
- The number of groups (combinations) of n different things taking any number at a time is:
nc0 + nc1 + nc2 +….. ncn = 2n – 1
- The number of ways (alternatively groups or combinations) in which n things can be divided into p things and q things is
- The number of ways (alternatively groups or combinations) in which 3n can be divided equally into three distinct groups each having n things is
- The number of ways (alternatively groups or combinations) in which 3n can be divided equally into three identical groups each having n things is
Back to Curriculum Guide
Table of contents
What is included in the year?
Overview of Grade 3 Major, Supporting, and Additional work
What is included in the year?
Grade 3 LearnZillion Math consists of 15 units and a total of 155 lessons. Each lesson is designed to be completed in one 45-55 minute class period. Each unit includes a summative Unit Assessment designed to be completed in one class period, this assessment day is not included in the number of days allotted to lessons. The organization of the units, and lessons within each unit, creates a coherent sequence based on the progressions of the standards. As shown in the figure below, the majority of lessons are focused on the major work of Grade 3. The remaining lessons focus on supporting or additional clusters, often in the service of the major work. In third grade, 72% of lessons (112 lessons) are devoted to the major work of the grade. Many lessons dedicated to major work standards are enhanced and deepened by simultaneously engaging students in standards from supporting or additional clusters. In some instances when appropriate, students work only with supporting or additional work clusters.
Grade 3 lesson focus across major, supporting, and additional clusters
Grade 3 units across the year
Overview of Grade 3 Major, Supporting, and Additional Work
Major work of the year
In third grade the major work focuses on the development of multiplicative reasoning. Students develop an understanding of the properties of multiplication and division, and the relationship between multiplication and division as they engage in explorations involving arrays, equal-sized groups, and area models. They develop strategies for solving multiplication and division problems within 100 and understand multiplication as finding an unknown product and division as finding an unknown factor. Third grade students become fluent with all the single digit combinations and also develop efficient strategies for calculating the products of whole numbers.
The major work with geometric measurement is connected to the development of multiplicative concepts. Third grade students are able to recognize area as an attribute of two-dimensional regions. They use the structure of a rectangular array to find the total number of same sized units, which will cover a specified shape. Building on the work of second grade with rectangular arrays and equal addends, third grade students now are able to connect multiplication to a rectangular array. They use the square to tile and justify that the area of a rectangle is the same as would be found by multiplying the side lengths. As students decompose and compose shapes, they begin to use area models to build ideas about the distributive property. Students also relate area to the operations of multiplication and addition.
Developing an understanding of fractions as numbers is a critical area in third grade that is foundational to ongoing major work in fourth and fifth grade. Students develop meaning for the unit fraction and can describe fractions as being built out of unit fractions. In first and second grade, students work with fractions as area models and that work is expanded in third grade as students compare fractions and use fractions to represent numbers equal to, less than, and greater than one. Representing fractions as a number and on a number line is introduced and students use the number line representation to justify equivalence and ordering of fractions. Students are able to compare fractions with the same numerator or same denominator by reasoning about the size of their size and by relating to the unit fraction. They also recognize that comparisons are only valid when the two fractions originate from the same sized wholes.
Third grade students solve a variety of problems involving the four operations and notice and explain arithmetic patterns. Students apply their understanding of the four operations and apply computation and reasoning strategies as they estimate and then solve measurement problems related to intervals of time, liquid volumes, and masses of objects.
The supporting work in third grade is central to the development of multiplicative ideas and the geometric measurement work. Students describe, compare, and analyze the properties of two-dimensional shapes. They use their understanding of the properties of shapes to classify shapes by their sides and angles. This angle work will support the upcoming related fourth grade standards. Throughout this work with shapes, third grade students can relate their fraction work to geometry by expressing the area of parts of shapes as a unit fraction of the whole.
Students further deepen their understanding of the operations as they represent and interpret data. As students use rulers to generate measurement data, they are applying major work concepts. They solve one and two step problems and represent their data finding with scaled graphs and line plots. Both of these representations of data support the work of multiplication and fractions.
The additional work of third grade is closely related to the major work standards. Students again work with the concept of geometric measurement as they solve problems about the perimeter of a plane shape. Students recognize perimeter as an attribute of plane figures and are able to differentiate between linear and area measures. Through a variety of experiences, such as measuring the perimeter of their classroom or desks and through solving real world problems such as constructing a pen for a classroom pet, students develop an understanding of the connections between perimeter and area; rectangles with the same perimeter can have different areas and rectangles with the same area can have different perimeters.
In third grade, students extend the meaning of the base ten structure as they explore what happens to a number when multiplied by a multiple of ten. They use place value understandings to round numbers to the nearest ten or hundred and then apply those strategies as an estimation method when solving problems. Students also reach fluency within 1000 when solving addition and subtraction problems. They develop efficient and generalizable methods using a range of strategies and algorithms for solving addition and subtraction problems.
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