Permutations And Combinations

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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:

, this is when there are no repetitions.

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

1. Karnataka text book for Class 10, Chapter 04-Permutations And Combinations

Additional Information

Useful websites

1. Permutation and combination mathsisfun
2. Permutation and combination themathpage

Useful video from khan academy and youtube

Reference Books

NCERT text book on permutations and combinations click here

Gujarat state text book on permutations and combinations click here

Teaching Outlines

Concept # 1 Fundamental Principle of Counting

Learning objectives

1. Students should be able to determine the number of outcomes in a problem
2. Students should be able to apply the Fundamental principle of counting to find out the total number of outcomes in problem
3. Students should be able to draw the tree diagram for the outcomes

Notes for teachers

when teaching permutation remember following points.

1. permutation is a way of representing number of outcomes in a event.

Activities

Activity No # 1 Flipping a coin and a dice Click here

Concept # 2 Factorial Notation

Learning objectives

1. Students should be able to use the factorial notation
2. Students should be able to tell that n! is the product of first 'n' natural numbers
3. Students should be able to know that if 'n' is a negative number or a decimal, n! is not defined
4. Students should be able to know the value of 0!

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.

Activities

Activity No # 1 For the activity to make students understand factorial notation please see the activity click here

Activity No # 2

Concept # 3 Permutations

Learning objectives

• Sudents should be able to state that permutation is an arrangement and write the meaning of ${^{n}}P_{r}$

• Sudents should be able to state that ${^{n}}P_{r}$=${\frac {n!}{(n-r)!}}$ and apply this to solve problems

• Sudents should be able to show that

1. ${(n+1)^{n}}P_{n}$=${^{n+1}}P_{n}$

1. ${^{n}}P_{r+1}$=${(n-r)^{n}}P_{r}$

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.

Activities

Activity No. 1 Permutation Activity 1

Activity No. 2 Create-a-Bear Permutations click here

Activity No. 3 Ice Cream Cone Permutations click here

Activity No. 4 Arranging books click here

Concept # 4 Combinations

Learning objectives

• State that a combination is a selection and write the meaning of ${^{n}}C_{r}$

• Distinguish between permutations and combinations

• Derive ${^{n}}C_{r}$=${\frac {n!}{(n-r)!r!}}$

and apply the result to solve problems

• Derive the relation ${^{n}}P_{r}$=${^{n}}C_{r}Xr!$

• Verify that ${^{n}}C_{n}$=${^{n}}C_{n-r}$ and give its interpretation

• Derive ${^{n}}C_{r}+^{n}C_{n-r}$=${^{n+1}}C_{r}$ and apply the result to solve problems.

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.

Activities

Activity No. 1 - Combination Activity 1

Activity No. 2 Be a Sport with Combinations click here

Activity No. 3 It's a Wrap with Combinations click here

Activity No. 4 Picking Books click here

Activity No. 5 - Permutation and Combination Activity 5

Assessment activities for CCE

Forming a kabbadi team click here

Further Explorations

A good page which describes the difference between permutations, combinations and how the formulae are arrived at, [[3]]

Hints for difficult problems

1.How many 3-digits numbers can be formed from the digits 0,1,2,3 and 4 without repetition?Solution

2.How many 4-digit numbers can be formed using the digits 1,2,3,7,8 and 9 (repetations not allowed)

1. How many of these are less than 6000?
   
2. How many of these are even?
   
3. How many of these end with 7? Solution
   

3.How many

1. lines
2. Triangles can be drawn through 8 points on a circle Solution

References

1. NCERT Mathematics Textbooks
2. Centre for Innovation in Mathematics Teaching. http://www.cimt.plymouth.ac.uk/menus/resources.htm