## Math 11 Chapter 2 Lesson 1: Counting Rules

## 1. Summary of theory

### 1.1. Addition rule

There are \(k\) options \({A_1},{A_2},{A_3},…,{A_k}\) to do the job. In there:

– There is \({n_1}\) how to do the option \({A_1}\),

– There is \({n_2}\) how to do the option \({A_2}\)

…

– There is \({n_k}\) how to do the option \({A_k}\).

Then the number of ways to do the job is: \({n_1} + {n_2} + … + {n_k}\) ways.

**Special:** If \(A\) and \(B\) are two non-intersecting finite sets, then the number of elements of \(A \cup B\) is equal to the sum of the elements of \(A\) and of \(B). \), ie: \(\left| {A \cup B} \right| = \left| A \right| + \left| B \right|\).

### 1.2. Multiplication rule

There are \(k\) stages \({A_1},{A_2},…,{A_k}\) to do the job.

– There is \({n_1}\) how to perform the step \({A_1}\).

– There is \({n_2}\) how to perform the step \({A_2}\).

…

– There is \({n_k}\) how to do the \({A_k}\) step.

Then the number of ways to do the job is: \({n_1}.{n_2}….. {n_k}\) ways.

## 2. Illustrated exercise

### 2.1. Exercise 1

Going from Hanoi to Ho Chi Minh City. Ho Chi Minh City can be reached by car, train, plane. Knowing that there are \(10\) cars, \(2\) trains and \(1\) flights that can enter the city. Ho Chi Minh. Number of ways to get into the city. Ho Chi Minh from Hanoi is:

**Solution guide**

There are \(3\) options to go from Hanoi to Ho Chi Minh City. Ho Chi Minh is: car, train, plane.

– There are \(10\) ways to go by car (because there are \(10\) trips).

– There are \(2\) ways to go by train (because there are \(2\) trips).

– There are \(1\) ways to go by plane (because there are \(1\) flights).

So there are all \(10 + 2 + 1 = 13\) ways to go from Hanoi and Ho Chi Minh City.

### 2.2. Exercise 2

Mai wants to set a house password with \(4\) digits. First digit is one of \(3\) digit \(1;2;0\), second digit is one of \(3\) digit \(6;4;3\), digit the third is one of \(4\) digits \(9;1;4;6\) and the fourth is one of \(4\) digits \(8;6;5;4\). How many ways for Mai to set a house password?

**Solution guide**

Setting a home password has \(4\) steps (from the first digit to the last digit).

– There is \(3\) how to do step 1 (corresponding to \(3\) how to choose the first digit).

– There is \(3\) how to do step 2 (corresponding to \(3\) how to choose the second digit).

– There is \(4\) how to do step 3 (corresponding to \(4\) how to choose the third digit).

– There is \(4\) how to do step 4 (corresponding to \(4\) how to choose the fourth digit).

So there are all \(3.3.4.4 = 144\) ways for Mai to set the home password.

### 2.3. Exercise 3

From the digits \(1, 2, 3, 4, 5, 6\) how many natural numbers less than \(100\) can be formed?

**Solution guide**

TH1: There are \(6\) 1-digit natural numbers made up of 1, 2, 3, 4, 5, 6.

TH2: From the digits \(1, 2, 3, 4, 5, 6\) make a two-digit natural number.

Call the two-digit natural number \(\overline {ab} \,\,\left( {a \ne 0} \right)\).

There are 6 ways to choose the digit a.

There are 6 ways to choose the digit b.

Apply the multiplication rule with \(6^2 = 36\) two-digit natural numbers made up of the digits 1, 2, 3, 4, 5, 6.

According to the addition rule there are \(6 + 36 = 42\) (number).

## 3. Practice

### 3.1. Essay exercises

**Question 1:** Nam went to the stationery store to buy a gift for a friend. There are three items in the store: Pens, notebooks and rulers, including \(5\) pens, \(4\) notebooks and \(3\) rulers. How many ways are there to choose a gift consisting of a pen, a ruler, and a notebook?

**Verse 2:** In a performance team, there are \(8\) male friends and \(6\) female friends. How many ways are there to choose a male-female duet?

**Question 3:** How many natural numbers have the property:

a) Is it even and has two digits (not necessarily different)?

b) Is it odd and has two digits (not necessarily different)?

c) Is it odd and has two different digits?

**Question 4: **In \(10000\) the first positive integer, how many numbers contain one digit \(3\), one digit \(4\) and one digit \(5\)?

### 3.2. Multiple choice exercises

**Question 1: **Arrange \(5\) students of class A and \(5\) students of class B into two rows of opposite seats, each row of \(5\) seats so that \(2\) students are facing each other different class. Then the number of ways to arrange is:

A. \(460000\)

B. \(460500\)

C. \(460800\)

D. \(460900\)

**Verse 2: **Use \(10\) digits \(0, 1, 2, 3, 4, 5, 6, 7, 8, 9\) to generate phone numbers with \(7\) digits. Then, the number of first phone numbers \(8\) is odd is:

A. \(5. {10}^5\)

B. \(5. {10}^6\)

C. \(2. {10}^6\)

D. \({10}^7\).

**Question 3: **Use \(10\) digits from \(0\) to \(9\) and \(26\) letters from A to Z to create a password consisting of \(6\) characters with at least one If the character is a letter, the password number is

A. \({26}^6-{10}^6\)

B. \({36}^6-{26}^6\)

C. \({36}^6-{10}^6\)

D. \({26}^6\)

**Verse 4. **A class has 23 girls and 17 boys. How many ways are there to choose a student to participate in the environmental competition?

A. 23

B. 17

C. 40

D. 391

**Question 5. **A bag contains 20 different marbles of which 7 are red, 8 are blue and 5 are yellow. The number of ways to get 3 marbles of different colors is

A. 20

B. 280

C. 6840

D. 1140

## 4. Conclusion

Through this lesson, you will learn some key topics as follows:

- Understand the definition of addition and multiplication rules.
- Master the skill of using counting rules and accurately calculate the number of elements each set is arranged according to a certain rule.

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