To apply some rule and product rule in solving problems.

To apply the principles of counting in solving problems.

*Lecture 9: Principles of Counting

*Activity 9.1

How many triangle can you draw using the 9 dots below as vertices?

*9.1 Sum RuleEveryday, there are 2 trains routine, 5 express bus routine, and 4 flight routine from Malaysia to Singapore. How many different ways can a passenger travel from Malaysia to Singapore?

Train

2

Bus5

Flight4

Total different

ways

11

*9.1 Sum RuleThe Sum Rule says that,

* If there are methods to complete a task.

*For the first method, , there are different ways to be done.

*For the Second method, , there are different ways to be done.

*For the method, , there are different ways to be done.

*For the last method, , there are different ways to be done.

The total different ways to complete the task

+

𝑤1

𝑤1

…

𝑤𝑛

Task

𝑀 2

*9.1 Sum Rule* In general, the Sum Rule can be written as

Number of total different ways a task can be done

*9.1 Sum ruleExample:

A student wants to borrow a book from library. He can choose the book from 3 business books, 5 computer science books, and 2 mathematics books. How many different ways he can borrow the book from library?

*3 + 5 + 2 = 10 different ways to borrow a books.

*Product rule*If John travel from town A to town C via

town B. There are 3 routes from town A to town B and 2 routes from town B to town C. In how many ways can John travel from town A to town C?

Town A

Town B

Town C

*Product Rule*If a task needs n steps to complete it.

Step 1 consists of ways, step 2 consists of ways, … ,and step n consists of ways, then the total different ways to complete the task is

Step 1, Step 2, … Step n,

Task

*Product rule*Jane has 5 different shirts and 4 different

jeans, how many different combination she can dress those shirts and jeans?

*Solution:Jane needs 2 steps to complete this task.Step 1: Choose a shirt. 5 different waysStep 2: Choose a pair of jean. 4 different ways

*Product ruleEach of five cards contain digit 0, 1, 2, 3, 4 respectively.

1. In how many ways these cards can be arranged to get an odd number?

2. In how many ways these cards can be arranged to obtain a number that is greater than 30,000.

3. In how many ways these cards can be arranged to obtain an odd number that is greater than 30,000?

*Product ruleExercise:

1. In how many ways can the word “Computing” can be arranged?

2. In how many ways can 3 persons be seated in an empty bus that has 44 seats.

Permutation and CombinationAssume that A, B, C are 3 students. 2 students are selected to take a photo. In how many ways we can arrange the 2 students?

Is AB and BA be considered as the same photo?

No.

Is AB and BA considered as the same team?

Yes.

When the order is important, the arrangement is called Permutation.

When the order is not important, the selection is called Combination.

AB AC CB

BA CA BC

PermutationIf r elements is to be arranged from n elements. The number of arrangements is

Example:

In how many ways can 4 out of 6 books be arranged in a shelf?

Solution:

6 books are available.

4 books are arranged.

CombinationIf r elements is to be selected from n elements. The number of Selection is

Example:

In how many ways can a team of 3 students be selected from 7 students?

Solution:

7 students are available.

3 students are selected.

*Activity 9.1

How many triangle can you draw using the 9 dots below as vertices?

CombinationTo form a triangle, we need to select 3 dots as vertices.

Therefore, 3 dots is selected from 9 dots.

Exercise:1. A person buying a personal computer system is

offered a choice of three models of the basic unit, two models of keyboard, and two models of printer. How many distinct systems can be purchased?

2. Suppose that a code consists of five characters, two letters followed by three digits. Find the number of:a) codes; b) codes with distinct letter.

3. Consider all positive integers with three digits. (Note that zero cannot be the first digit.) Find the number of them which are: a) greater than 700; b) odd;c) divisible by 5.