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Permutations And Combinations (view source)
Revision as of 11:49, 30 January 2020
, 11:49, 30 January 2020→Hints for difficult problems
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[http://www.karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_History The Story of Mathematics]
[http://www.karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_History The Story of Mathematics]
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|[http://www.karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Philosophy Philosophy of Mathematics]
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[http://www.karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Pedagogy Teaching of Mathematics]
[http://www.karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Pedagogy Teaching of Mathematics]
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[http://www.karnatakaeducation.org.in/KOER/en/index.php/Maths:_Curriculum_and_Syllabus Curriculum and Syllabus]
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[http://www.karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Topics Topics in School Mathematics]
[http://www.karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Topics Topics in School Mathematics]
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[http://www.karnatakaeducation.org.in/KOER/en/index.php/Text_Books#Mathematics_-_Textbooks Textbooks]
[http://www.karnatakaeducation.org.in/KOER/en/index.php/Text_Books#Mathematics_-_Textbooks Textbooks]
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[http://www.karnatakaeducation.org.in/KOER/en/index.php/Maths:_Question_Papers Question Bank]
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= Concept Map =
= Concept Map =
[[File:Permutation_and_Combinations.mm|Flash]]
__FORCETOC__
__FORCETOC__
= Permutations & Combinations =
== Fundamental Counting Principle ==
If an activity can be done in ‘m’ ways and other activity can be done in ‘n’ ways, the total number of ways of doing the activity one after the other is '''m X n'''
'''Illustration'''--- Ramu can go from his home to school by 3 means --- by walk,
by cycle, by bus to near school bus stand, from there he reach the
school by 2 ways (kacha road, pakka road).
walk kacha Home cycle School Bus stand School bus pakka
3 ways from home to school stand
2 ways from stand to school
m x n = 3 x 2 = 6 ways
'''Examples using the counting principle:''' Let's say that you want to flip a coin and roll a die. There are 2 ways that you can flip a coin and 6 ways that you can roll a die. There are then 2x6=12 ways that you can flip a coin ''and'' roll a die.
If you want to draw 2 cards from a standard deck of 52 cards without
replacing them, then there are 52 ways to draw the first and 51 ways
to draw the second, so there are a total of 52*51 = 2652 ways to draw
the two cards.
== Permutations ==
A permutation is one of the different arrangements of a group of items where order matters.
Consider the following:
Given 3 people, Brinda, Mary and Shariff, how many different ways can these three people be arranged where order
matters?
Let BMS stand for the order of Brinda on the left, Mary in the middle and Shariff on the right.
Since order matters, a different arrangement is BSM. Where Brinda is on the left, Shariff is in the middle and Mary is on the right.
If we find all possible arrangements of Brinda, Mary and Shariff where order matters, we have the following:
BMS, BSM, MSB, MBS, SMB, SBM
The number of ways to arrange three people three
at a time is: ''' 3! = (3)(2)(1) = 6''' ways
There are n! ways to arrange n
objects in groups of n at a time.
The number of permutations of n objects taken r at
a time is:
[[Image:KOER%20Probability,%20Permutations%20and%20Combinations_html_5d806af0.gif]],
this is when there are <u>no repetitions</u>.
== Combinations ==
A combination is of a grouping of items where order does not matter. Example creating subgroups of 3 from Sub groups of 3 from a team of fifteen.
Consider the following:
Given 3 people, Brinda, Mary and Shariff, how many different ways can these three people be selected to form a group?
Since order does not matter, any arrangement with Brinda, Mary and Shariff is considered the same .
Therefore the only one way of grouping 3 people.
Now suppose we want to take four people, Brinda, Mary, Shariff and Anita, and arrange them in groups of three at a
time where order does not matter. The following demonstrates all the possible arrangements.
BMS, MSA, BMA, BSA
There are 4 ways to group 4 people in groups of 3 at a time.
The number of combinations of a group of n objects taken r at a time is:
when no repetitions are allowed.
= Textbook =
= Textbook =
#[http://ktbs.kar.nic.in/New/Textbooks/class-x/english/maths/class-x-english-maths-chapter04.pdf Karnataka text book for Class 10, Chapter 04-Permutations And Combinations]
=Additional Information=
=Additional Information=
==Useful websites==
==Useful websites==
#mathsisfun[http://www.mathsisfun.com/combinatorics/combinations-permutations.html Permutation and combination]
#[http://www.mathsisfun.com/combinatorics/combinations-permutations.html Permutation and combination mathsisfun]
#themathpage[http://www.themathpage.com/aprecalc/permutations-combinations.html Permutation and combination]
#[http://www.themathpage.com/aprecalc/permutations-combinations.html Permutation and combination themathpage]
'''Useful video from khan academy and youtube'''
{{#widget:YouTube|id=oQpKtm5TtxU}}
{{#widget:YouTube|id=XqQTXW7XfYA}}
{{#widget:YouTube|id=v9NLtiVt3XY}}
==Reference Books==
==Reference Books==
===Notes for teachers===
===Notes for teachers===
''These are short notes that the teacher wants to share about the concept, any locally relevant information, specific instructions on what kind of methodology used and common misconceptions/mistakes.''
'''when teaching permutation remember following points.'''
# permutation is a way of representing number of outcomes in a event.
===Activities===
===Activities===
'''Activity No # 1''' Flipping a coin and a dice
'''Activity No # 1''' Flipping a coin and a dice
[[Permutations_And_Combinations_Fundamental_principle_of_counting_Activity1|Click here]]
[[Permutations_And_Combinations_Fundamental_principle_of_counting_Activity1|Click here]]
==Concept # 2 Factorial Notation==
==Concept # 2 Factorial Notation==
==Concept # 3 Permutations==
==Concept # 3 Permutations==
===Learning objectives===
===Learning objectives===
*State that permutation is an arrangement and write the meaning of <math>{^{n}}P_{r}</math>
*Sudents should be able to state that permutation is an arrangement and write the meaning of <math>{^{n}}P_{r}</math>
<br>
<br>
*State that <math>{^{n}}P_{r}</math>=<math>\frac{n!}{(n-r)!}</math> and apply this to solve problems
*Sudents should be able to state that <math>{^{n}}P_{r}</math>=<math>\frac{n!}{(n-r)!}</math> and apply this to solve problems
<br>
<br>
*Show that
*Sudents should be able to show that
<br>
<br>
#<math>{(n+1)^{n}}P_{n}</math>=<math>{^{n+1}}P_{n}</math>
#<math>{(n+1)^{n}}P_{n}</math>=<math>{^{n+1}}P_{n}</math>
===Activities===
===Activities===
'''Activity No # 1 Create-a-Bear Permutations'''
'''Activity No. 1 [[Permutation Activity 1]]'''
'''Activity No. 2 Create-a-Bear Permutations'''
[[Permutations_And_Combinations_permutations_activity1|click here]]
[[Permutations_And_Combinations_permutations_activity1|click here]]
'''Activity No # 2 Ice Cream Cone Permutations'''
'''Activity No. 3 Ice Cream Cone Permutations'''
[[Permutations_And_Combinations_permutations_activity2|click here]]
[[Permutations_And_Combinations_permutations_activity2|click here]]
'''Activity No # 3 Arranging books'''
'''Activity No. 4 Arranging books'''
[[Permutations_And_Combinations_permutations_activity3|click here]]
[[Permutations_And_Combinations_permutations_activity3|click here]]
===Activities===
===Activities===
'''#Activity No # 1 Be a Sport with Combinations'''
'''Activity No. 1 - [[Combination Activity 1]]'''
[[Permutations_And_Combinations_combinations_activity1|click to see]]
'''Activity No. 2 Be a Sport with Combinations'''
[[Permutations_And_Combinations_combinations_activity1|click here]]
'''#Activity No #2 It's a Wrap with Combinations'''
'''Activity No. 3 It's a Wrap with Combinations'''
[[Permutations_And_Combinations_combinations_activity2|click to see]]
[[Permutations_And_Combinations_combinations_activity2|click here]]
'''#Activity No #3 Picking Books'''
'''Activity No. 4 Picking Books'''
[[Permutations_And_Combinations_combinations_activity3|click to see]]
[[Permutations_And_Combinations_combinations_activity3|click here]]
'''Activity No. 5 -''' [[Permutation and Combination Activity 5]]
=Assessment activities for CCE=
=Assessment activities for CCE=
Forming a kabbadi team [[Permutations_and_combinations_assessment_activities_for_CCE| click here]]
== Further Explorations ==
A good page which describes the difference between permutations, combinations and how the formulae are arrived at, [[http://www.mathsisfun.com/combinatorics/combinations-permutations.html <nowiki>[3]</nowiki>]]
= Hints for difficult problems =
= Hints for difficult problems =
'''1.How many 3-digits numbers can be formed from the digits 0,1,2,3 and 4 without repetition?'''[[Class10_permutations_and_combinations_problems#Problem_1|Solution]]
'''1.How many 3-digits numbers can be formed from the digits 0,1,2,3 and 4 without repetition?'''[[Class10_permutations_and_combinations_problems#Problem_1|Solution]]
<br>
<br><br>'''2.How many 4-digit numbers can be formed using the digits 1,2,3,7,8 and 9 (repetations not allowed)'''<br>
<br>
'''2.How many 4-digit numbers can be formed using the digits 1,2,3,7,8 and 9 (repetations not allowed)'''<br>
#How many of these are less than 6000?<br>
#How many of these are less than 6000?<br>
#How many of these are even?<br>
#How many of these are even?<br>
#How many of these end with 7? [[Class10_permutations_and_combinations_problems#Problem_2|Solution]]<br>
#How many of these end with 7? [[Class10_permutations_and_combinations_problems#Problem_2|Solution]]<br>
'''3.How many'''
'''3.How many'''
= Project Ideas =
= Project Ideas =
= Math Fun =
= References =
# NCERT Mathematics Textbooks
# Centre for Innovation in Mathematics Teaching. http://www.cimt.plymouth.ac.uk/menus/resources.htm
[[Category:Permutations and Combinations]]
