Permutations.docx

Lesson Plan in Mathematics 10 I.

II. III.

IV.

Weekly Objectives A. Content Standards The learner demonstrates understanding of key concepts of combinatorics and probability. B. Performance Standards The learner is able to use precise counting technique and probability in formulating conclusions and making decisions C. Learning Competencies/Objectives 1. Illustrates the permutation of objects. 2. Develop skills in counting techniques . 3. Participate actively in class activity and discussion Concept: Statistics and Probability Learning Resources A. References : ( Teachers guide pp 285 and 310 ) B. Other Learning Resources Internet, Google, Youtube Procedures A. Preliminary Activities 1. Prayer 2. Greetings 3. Presentation of the Objectives of the Lesson B. Establishing a purpose for the lesson Teacher’s Activity

Learner’s Activity

Lets ROLL It So I have here a die and I’m going to roll it. So what is the probability of getting any single number on it ? Do you have an idea, Yes Ellaine? Very Good, The probability of getting any single number from rolling a six sided die is 1/6. Since the sample space contains the value of S = { 1, 2, 3 ,4, 5 and 6 } suppose I’m going to pick 2 number from it say event A = ( 3,4 ) What is the probability of getting this number ?

It is 2/6 or 1/3

Yes Marichu? Very Good Marichu! For instance to get the probability of an event it simply 𝒏( 𝑨 )

The probability is 1/6

𝟐

P(A) = 𝒏( 𝑺 ) , P(A) = 𝟔 or

𝟏 𝟑

C. Presenting the new lesson

1. Probability

2. Factorial

3. Permutations

permutation is an arrangement of all or part of a set of objects with regards to the order of the arrangement.

D. Presenting new lesson / instances of the new lesson But what if I’m going to roll 2 dice ? What would be my sample space now ? So if I am going to roll this 2 dice my sample space would include the value of S = { 1,1 },{ 1,2 },{ 1,3 }…,{ 1,6 } { 2,1 },{ 2,2 },{ 2,3 }…,{ 2,6 } . . { 6,1 },{ 6,2 },{ 6,3 }…,{ 6,6 } What if I have 3 or 4 or 5 dice. What would be my sample space ? Do we need to list every single elements ? My point here is that it is possible for us to have an experiment having a sample space that contains hundredth of elements or thousandth or even millions. Listing every elements of sample space will take us a lot of time. But Math has an easy way on how we are going to deal with this kind of problem. One way to solve it is using the Multiplication rule of counting. For example, Let say that You are eating in a small restaurant that offers only 3 appetizers and 2 main dishes. In how many ways you can order your meal ?

So… Fig. 1

Over all there are 6 possible ways of choosing your meal. The number of ways of choosing your meal is equal to the total number of appetizers ( 3 ) x total number of dishes ( 2 ) In general If you have an experiment which is going to be done in rth different stages That is… Stage 1 – Stage 2 – Stage 3 … – Stage r Also if you have 𝑛1 of doing Stage 1, 𝑛2 of doing Stage 2 up to 𝑛𝑟 of doing the Stage r, That is… Stage 1 – Stage 2 – Stage 3 … – Stage r 𝒏𝟏 𝒏𝟐 𝒏𝟑 … 𝒏𝒓 Then the number of ways of doing the experiment is 𝒏𝟏 𝒙 𝒏𝟐 𝒙 𝒏𝟑 … 𝒙 𝒏𝒓 And that is the Multiplication Rule of Counting Is that clear ? Yes sir… Let us say I have an urn here containing 4 distinct marbles, ( red, blue, green and black ) and I’m going to conduct an experiment. That is I’m going to draw a marbles then after knowing its color I will return it to the urn, Then I’m going to draw a marble again, after knowing its color I will put it back again to the same urn. In how many ways I can do it ?

Fig. 2

1st Draw

2nd Draw

So in 1st draw I have 4 choices of marbles and in 2nd draw I still have 4 choices, since I return the marble I have drawn to the same urn. So it is simply 4 x 4 = 16 But what if I’m going to make some changes in my sampling schemes. I will do the same experiment but this time without replacing back the marbles to the urn. Fig. 3

1st Draw

2nd Draw

Let’s say I picked the color green on my 1st draw, since I will not put it back to the urn, in my 2nd draw I now only have 3 choices of marbles In this case.. 4 x 3 = 12 ways So we might ask a question of, In how many ways can a series of an rth distinct object can draw from nth distinct object?

\

In that case the number of ways of doing that is equal to n

1

st

n-1

n-2

nd

rd

2

3

........... ............

n–r+1

rth

( n ) ( n -1 ) ( n – 2 ) … ( n – r + 1 ) When we do this kind of sampling we call this as Permutation of rth distinct objects from nth distinct objects where ( n ≥ r ) Understood ?

Yes sir…

We use a symbol for this: nPr = ( n ) ( n -1 ) ( n – 2 ) … ( n – r + 1 ) x ( n – r )( n – r – 1 ) … 3 x 2 x 1 ( n – r )( n – r – 1 ) … 3 x 2 x 1

Which is equal to : nPr = (

𝒏! 𝒏−𝒓 )!

where n! = ( n – r )( n – r – 1 ) … 3 x 2 x 1 For example : 1. Suppose there 10 athletes competing in 100 meters run. In how many ways can the runners arrange their ranking in top 3 competitors. 10 9 8 st nd 1 2 3rd st So in 1 place there have 10 options, in 2nd place there have only 9 since one runner already crossed the finish line and in 3rd place there have 8 options of runners because 2 runners crossed already the finish line. So the number of ways of ranking the runners to top 3 competitors is 10 x 9 x 8 = 720 One thing I want you to note here is that Ordering Matters 10P3 = (

10! 10−7 )!

=

10! 7!

=

10 𝑥 9 𝑥 8 𝑥 7 𝑥 6 𝑥 5 𝑥 4 𝑥 3𝑥 2 𝑥 1 7𝑥6𝑥5𝑥4𝑥3𝑥2𝑥1

= 720 ways So why do I say order matters ? Let say Carl came first next is Harold and next is Carlo. This order is completely different if Carlo came first next is Harold and next is Carl. That’s why ordering matters in permutation.

Understood ? Yes sir…

Let’s say if I’m going to get the permutation of n objects from n objects ( nPn ), I’m pointing the orderings of then distinct objects. Example: Lets say there are 5 distinct books ( Math1, Math2, Math3, Math4, English4 ) in how many ways can I arrange it in the bookshelf ? So it is equal to : 5P5 =

5! 5! = ( 5−5 )! 0!

= 5! =

Note that 0! = 1 Is that clear ? Yes sir…

E. Mastery of the content

Identify the following situations that illustrate Permutations 1. Selecting 5 noisy classmates in your class. 2. Forming a group consisting of 3 members from 6 people. 3. 5 people arranging themselves in a row for selfie. 4. 7 people arranging themselves to sit in a bus having only 3 available sits. 5. Electing your classmates as President, Vice President, and Secretary in your classroom. 6. Arranging the 4 letters word L O V E 7. Determining your top 3 crushes in the school. 8. Selecting 4 team mates in rank mode in MOBILE LEGENDS. 9. Choosing 3 of your classmates to attend in your birthday party. 10. Picking 5 friends to chat in your 99+ online.

The situations that illustrate permutations is item: 2. Forming a group consisting of 3 members from 6 people. 3. 5 people arranging themselves in a row for selfie. 4. 7 people arranging themselves to sit in a bus having only 3 available sits. 5. Electing your classmates as President, Vice President, and Secretary in your classroom. 6. Arranging the 4 letters word L O V E 7. Determining your top 3 crushes in the school.

F. Finding practical applications of concepts and skills in daily living

So this time I want you to group into 5 groups. Each group will going pick situations and explain or illustrate why that situation considered as permutation

Students will present their explanation in front

Pick a representative to explain here in front or illustrate the situation by group.

G. Making generalizations and abstractions about the lesson

So let’s make a recap about our discussion Again what is the formula we use in finding the permutation of rth distinct object from nth distinct objects ? Yes.. Kristel ? Very good Kristel

𝒏!

Sir it’s nPr = ( 𝒏−𝒓 )!

𝒏!

We use nPr = ( 𝒏−𝒓 )! How about if I ask the permutation of nth distinct objects from nth distinct objects ? Yes Angelica ? That’s right angelica

Its n! sir

It is n! Does order matters in permutation ? Very good ! ordering is matter in permutations.

Yes..

H. Evaluating learning Answer the following: 1. In how many ways can you arranged the letters in the word ROCES ? 2. If you’re going to arrange numbers { 1 2 3 … 9 ) four at a time without replacement in how many ways you can do it ? 3. If you have 3 coins and you are going to toss it all together, In how many Possible ways could it appear ?

I. Assignment In your note book answer the following stated situations that describe permutations. Answer completely and neatly. 1. 5 people arranging themselves in a row for selfie. 2. 7 people arranging themselves to sit in a bus having only 3 available sits. 3. Electing your classmates as President, Vice President, and Secretary in your classroom. 4. Arranging the 4 letters word L O V E 5. Determining your top 3 crushes in the school.