permutations lesson

Permutation Revision

Permutations refer to the different possible arrangements of a set of objects.

Guide Question :

How does the concept of permutation help in forming conclusions and in making wise decisions?

Suppose you are going to see your friends at Sampaloc Lake because you are going to spend one-hour bonding with them through biking. Sampaloc Lake is the most popular lake among the Seven Lakes in San Pablo City.

Many tourists are visiting this place and biking is one of the most well- liked recreations by teen -agers like you. 

But you have a problem about the suit that you are going to wear. You need to make a decision by choosing among your 5 shirts, 3 pants, 2 rubber shoes and 4 socks.

In how many ways are you going to arrange them in order for you to decide which seems to be the perfect in your taste?

Let us have illustrations by arranging them using the Tree Diagram.

You have to match the shirts, the pants, the shoes and the socks in different ways. Then, find out how many ways you can possibly arrange them. By using the Tree Diagram , you are able to find out the answer : 120 ways. 

We can also do this by Systematic Listing:

white shirts, maong pants, adidas, stripes socks

blue shirts, maong pants, adidas, stripes socks

red shirts, maong pants, adidas, stripes socks

green shirts, maong pants, adidas, stripes socks

yellow shirts, maong pants, adidas, stripes socks

Fundamental Counting Principle 

If there are m ways to do one thing , n ways to do another, o ways to do another and p ways to do another, then , there are m x n x o x p of doing those things.

Fill in the Blank

Therefore, in our example if we use FCP we just multiply : (5)(3)(2)(4) =

During Fiesta, as one of our traditions, sweet delicacies are always present. Your mother prepares three types of these: Ubeng Halaya, Buko Salad, and Sweetened Macapuno.

If you are supposed to help your mother in preparing the dishes to be served to your visitors, then, in how many possible ways can you serve the three sweet delicacies?

By Systematic Listing

 

 

Ubeng Halaya,Buko Salad, Macapuno

Ubeng Halaya, Macapuno, Buko Salad

Buko Salad, Ubeng Halaya, Macapuno

Buko Salad,Macapuno, Ubeng Halaya

Macapuno, Ubeng Halaya, Buko Salad

Macapuno, Buko Salad, Ubeng Halaya

As you can see from the Tree Diagram and Systematics Listing , There are 6 possible ways that you can serve the sweet delicacies.

Using the FCP(Fundamental Counting Principle

We have : m x n x o = (3)(2)(1) = 6

6 possible ways of serving the sweet delicacies

In this example you notice that the factors are decreasing. Another way of writing (3)(2)(1) is 3! ( read as 3 factorial ).

Therefore, 3! = (3)(2)(1) = 6 ;  3! = 6 

Permutations taken all at a time

Permutations taken r at time

Distinguishable Permutations

Circular Permutations

​Factorial of natural number n (Giai thừa của số tự nhiên n) is denoted as:

• 0! = 1.

• n! = 1 x 2 x 3 x ... x n.

• Giai thừa của n là số cách xếp n người vào n cái ghế trên một hàng.

​1! = 1.

2! = 1 × 2 = 2

3! = 1 ×2 × 3 = 6

4! = 1 × 2 × 3 × 4 = 24

5! = 1 × 2 × 3 × 4 × 5 = 120

6! = 1 × 2 × 3 × 4 × 5 × 6 = 720

7! = ​1 × 2 × 3 × 4 × 5 × 6 × 7 = 5040

n! = (n-1)! × ​N.

Multiple Choice

$$\frac{10!}{8!}\ =\$$  

• 80

• 9

• 10

• 90

Multiple Choice

$$4!=...$$  

• $$4+3+2+1$$  

• $$1x2x3x4$$  

• $$4+5+6$$  

• $$4$$  

Multiple Choice

$$\frac{100!}{101!}\ =\$$  

• 101

• 1/100

• 100

• 1/101

Permutation: is the number of ways we can arrange k people of the group of n people in a row of k chairs (k ≤ n), such that no person hold more than one chair. We denote it as: P(n,k)

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Hoán vị (hoán đổi vị trí) là số cách xếp k người từ nhóm n người vào k chiếc ghế trên một hàng (k ≤ n). Ta ký hiệu là: P(n,k).​

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Ví dụ 1: Có bao nhiêu cách xếp 1 người từ nhóm 5 người vào 1 cái ghế?

P(5,1) = 5​

​

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Multiple Choice

How many ways can a club select a president, vice president, and a secretary from a group of 6 people?

Có bao nhiêu cách để một câu lạc bộ chọn được chủ tịch, phó chủ tịch và thư ký từ một nhóm 6 người?

• 30

• 120

• 720

• 240

Multiple Choice

There are 6 books of Math, Physics, Chemistry, Biology, English, and France are arranged on 1 row of bookshelf. How many ways are there to arrange the Math book on the left side of the Biology book?

• $$240$$  

• $$360$$  

• $$120$$  

• $$720$$   

Multiple Choice

In how many ways can 3 people can be seated around a round table?

• $$6$$  

• $$2$$  

• $$3$$  

• $$1$$