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## Solve Exercises Lesson 1. Addition rules. Multiplication rule. Math tree diagram (C5 – Math 10 Kite)

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### Solution of Exercises Lesson 1, page 9 of Math textbook 10 Kite, episode 2

From the digits 1, 2, 3, 4, 5, 6, we form a three-digit natural number that is divisible by 5. How many such numbers can be made?

**Solution method**

Perform consecutive actions to select the units, tens, and hundreds digits.

**Solution guide**

Creating a three-digit natural number that is divisible by 5 is to perform 3 consecutive actions: select the units digit, select the tens digit, and select the hundreds digit.

select the unit digit: There is 1 way to choose (number 5).

choose tens digit: There are 6 ways to choose.

select hundreds digit: There are 6 ways to choose.

According to the multiplication rule, the number of natural numbers that can be formed is: 1.6.6=36 (number).

### Solve the exercises Exercise 2 page 10 Math textbook 10 Kite episode 2

From the numbers 1, 2, 3, 4, 5, 6, 7, how many can you make?

a) Three-digit even number?

b) Even numbers consisting of three distinct double digits?

**Solution method**

Perform consecutive actions: select units, tens, hundreds digits.

**Solution guide**

a) Making a three-digit even number is performing 3 actions in a row: selecting the unit digit, choosing the tens digit, and choosing the hundreds digit.

select unit digit: There are 3 ways to choose (number 2, 4, 6).

choose tens digit: There are 7 ways to choose.

select hundreds digit: There are 7 ways to choose.

According to the multiplication rule, the number of even numbers that can be made is: 3.7.7=147 (number).

b) Making an even number consisting of three different double digits is to perform 3 consecutive actions: select the unit digit, select the tens digit, and select the hundreds digit.

select unit digit: There are 3 ways to choose (number 2, 4, 6).

choose tens digit: There are 6 ways to choose.

select hundreds digit: There are 5 ways to choose.

According to the multiplication rule, the number of even numbers that can be made is: 3.6.5=90 (number).

### Solve the exercises Lesson 3 page 10 Math textbook 10 Kite episode 2

In a high school, grade 10 has 245 boys and 235 girls.

a) The school needs to choose a student in grade 10 to attend an exchange with students from high schools in the province. Ask the school how many ways to choose?

b) The school needs to choose two students in grade 10, including 1 male and 1 female, to attend a student’s summer camp in the province. Ask the school how many ways to choose?

**Solution method**

a) Take one of two actions: select a male student or select a female student.

b) Perform two consecutive actions: select a male student, select a female student.

**Solution guide**

a) The selection of a student to attend the exchange is to do one of the following two activities:

Choose a male student: There are 245 ways to choose.

Choose a female student: There are 235 ways to choose.

So there are 245 +235 ways to choose a student to attend the exchange.

b) The selection of two students to attend the summer camp requires the following two consecutive activities:

Choose a male student: There are 245 ways to choose.

Choose a female student: There are 235 ways to choose.

So there are 245,235=57575 ways to choose two students to go to summer camp.

**Attention**

Question b: we can change the order of execution: choose a female student, then choose a male student.

### Solve the exercises Lesson 4, page 10, Math textbook 10, Kite episode 2

In the World Cup football tournament, the group stage has 32 participating teams, divided into 8 groups, each group has 4 teams playing in a round robin. Calculate the number of matches played in the group stage according to the above formula.

**Solution method**

Calculate the number of matches in each table according to the multiplication rule.

**Solution guide**

For each group, the symbols for 4 teams are A, B, C, D respectively.

The number of matches is the number of ways to select 2 teams to compete in the table, performing the following activities consecutively:

Choose a team to play against team A: There are 3 ways to choose

Choose a team to play against team B: There are 2 ways to choose

Choose a team to play against team C: There is 1 way to choose

So there will be 3.2.1 = 6 matches in each group.

So 8 groups have: 8.6 = 48 matches played in the group stage

**Attention:**

The format of the round-robin competition is: each team will meet all the other teams in the group in turn, playing only once.

### Solve exercises Exercise 5 page 10 Math textbook 10 Kite episode 2

In Canada, postal codes have 6 characters: 3 uppercase letters (out of 26 English letters) and 3 digits. Each postal code begins with a letter and alternates with a number.

(Source: https://capath.vn/postal-code-canada)

a) How many postal codes can be generated?

b) How many codes starting with the letter S can be generated?

c) How many codes can be generated starting with the letter S and ending with the number 8?

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**Solution method**

a) Select the characters one by one and then apply the multiplication rule

b) Step 1: Choose the first character “S” in 26 capital letters

Step 2: Select the next 5 characters in turn

Step 3: Apply the multiplication rule

c) Step 1: Choose the first character “S” in 26 capital letters

Step 2: Select the last character as the digit “8” in 10 digits

Step 3: Choose the remaining 4 characters in turn

Step 4: Apply the multiplication rule

**Solution guide**

a) +) Number of ways to choose the first character from the alphabet is: 26 (way)

+) Number of ways to choose the second character from 10 digits is: 10 (way)

+) Number of ways to choose the third letter from the alphabet is: 26 (way)

+) Number of ways to choose the fourth character from 10 digits is: 10 (way)

+) Number of ways to choose the fifth letter from the alphabet is: 26 (way)

+) Number of ways to choose the last character from 10 digits is: 10 (way)

+) Applying the multiplication rule, we have the number of postal codes that can be generated as: (postal code)

b) +) Since the first character to be selected is the letter “S”, the number of ways to choose the first character is: 1 (way)

+) Number of ways to choose the second character from 10 digits is: 10 (way)

+) Number of ways to choose the third letter from the alphabet is: 26 (way)

+) Number of ways to choose the fourth character from 10 digits is: 10 (way)

+) Number of ways to choose the fifth letter from the alphabet is: 26 (way)

+) Number of ways to choose the last character from 10 digits is: 10 (way)

+) Applying the multiplication rule, we have a postal code that can be generated as: (way)

c) +) Since the first character to be selected is the letter “S”, the number of ways to choose the first character is: 1 (way)

+) Number of ways to choose the second character from 10 digits is: 10 (way)

+) Number of ways to choose the third letter from the alphabet is: 26 (way)

+) Number of ways to choose the fourth character from 10 digits is: 10 (way)

+) Number of ways to choose the fifth letter from the alphabet is: 26 (way)

+) Since the last character to be selected is the digit “8”, the number of ways to choose the last character is: 1 (way)

+) Applying the multiplication rule, we have a postal code that can be generated as: (way)

### Solving exercises Lesson 6 page 10 Math textbook 10 Kite episode 2

A fashion company launched a new shirt model with three colors: white, blue, black. Each type has sizes S, M, L, XL, XXL.

a) Draw a tree diagram showing the types of shirts with the color and size mentioned above.

b) If a store wants to buy all kinds of shirts (all kinds of colors and all sizes) and one of each type to introduce, how many shirts should be purchased in all?

**Solution method**

a) Draw a tree diagram in the order of choosing: Shirt color – Shirt size.

b) Based on the tree diagram, count the number of shirts to buy.

**Solution guide**

b) If a store wants to buy all kinds of shirts (all colors and sizes) and one of each type to introduce, it needs to buy all 15 shirts.

### Solve exercises Exercise 7 page 10 Math textbook 10 Kite episode 2

A small hotel prepares a breakfast consisting of 2 drinks: tea and coffee; 3 dishes are: pho, vermicelli and porridge; The 2 desserts are: banh cuon and sourdough.

a) Draw a tree diagram showing how to choose a diet consisting of all three types: drinks, meals and desserts.

b) Calculate the number of ways to choose ceramic servings: 1 drink, 1 dish and l dessert.

**Solution method**

a) Draw a tree diagram in the order of choosing: Drinks – Food – Desserts.

b) Based on the tree diagram, we count the number of choices.

**Solution guide**

b) Based on the tree diagram, we have the number of ways to choose a serving including: 1 drink, 1 dish and l dessert: 12 (choices)

### Solve exercises Exercise 8 page 10 Math textbook 10 Kite episode 2

Give genotype AaBbDdEe. Assuming normal meiosis, no mutation occurs.

a) A tree-shaped plot showing gamete formation.

b) From there, calculate the number of crossings of genotype AaBbDdEe.

**Solution method**

a) Draw a tree diagram

b) Based on the tree diagram, we count the number of choices.

**Solution guide**

a) Tree diagram showing gamete formation:

b) Based on the tree diagram, the number of crossings from genotype AaBbDdEe is: 16 (types)