You are currently viewing Solve the exercises at the end of chapter 5 – Math 10 Kite – Math Book

Solve the exercises at the end of chapter 5 – Math 10 Kite – Math Book

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Solve the exercises at the end of chapter 5 – Math 10 Kite
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Solve the exercises Exercise 1 page 20 Math textbook 10 Kite episode 2

a) How many ways are there to arrange 20 students in a vertical line?

A.\({20^{20}}\) B.\(20!\) C. 20 D.1

b) The number of ways to select 3 students from a class of 40 students is:

A. \(A_{40}^3\) B. \({40^3}\) C. \({3^{40}}\) D.\(C_{40}^3\)

Solution method

a) Line up 20 students in a vertical line \( \Rightarrow \) Use the permutation formula

b) Select 3 students from a class of 40 students \( \Rightarrow \) Use combinatorial formula

Solution guide

a) The number of ways to arrange 20 students in a vertical line is: \(20!\) (order). So we choose answer B.

b) The number of ways to select 3 students from a class of 40 students is: \(C_{40}^3\) (choice). So we choose answer D.

Solve the exercises Exercise 2 page 20 Math textbook 10 Kite episode 2

Duong has 2 pairs of pants including: one blue pants and one black pants; 3 shirts include: a brown shirt, a blue shirt and a yellow shirt, 2 pairs of shoes including: a pair of black shoes and a pair of red shoes. Duong wants to choose a set of clothes and a pair of shoes to go sightseeing. By drawing a tree diagram, calculate the number of ways to choose an outfit and a pair of shoes for Duong.

Solution method

Draw a tree diagram by selecting the pants first, then the shirt, and finally the shoes. Then count the number of ways to choose.

Solution guide

Solve the exercises at the end of chapter 5 – Math 10 Kite 2

Conclusion: From the tree diagram, we see that Duong has 12 ways to choose a set of clothes and a pair of shoes.

Solve the exercise Exercise 3 page 20 Math textbook 10 Kite episode 2

In the plane, give two parallel lines a and b. Given 3 distinct points on line a and 4 distinct points on line b. How many triangles have all 3 vertices at 3 of these 7 points?

Solution method

A triangle is made up of 3 non-collinear points, so to have a triangle we will choose 3 non-collinear points out of 7 given points.

Method 1:

Take 2 points in a, 1 point in b and vice versa

Method 2:

Calculate the number of ways to choose any 3 points from 7 points – the number of ways to choose 3 collinear points on a and b.

Solution guide

Method 1:

TH1: 2 points belong to a and 1 point belongs to b

Number of ways to choose 2 points on line a is \(C_3^2\) (how to choose)

The number of ways to choose a point on line b is: \(C_4^1\) (how to choose)

=> The number of triangles formed is: \(C_3^2 . C_4^1 = 12\)

TH2: 2 points in b and 1 point in a

Number of ways to choose 2 points on line b is \(C_4^2\) (how to choose)

The number of ways to choose a point on the line a is: \(C_3^1\) (how to choose)

=> The number of triangles formed is: \(C_4^2 + C_3^1 = 18\)

So there are 12 + 18 = 30 triangles in all.

Method 2:

Number of ways to choose 3 points on line a is: \(C_3^3\) (how to choose)

Number of ways to choose 3 points on line b is: \(C_4^3\) (how to choose)

Number of ways to choose any 3 points out of 7 given points is: \(C_7^3\) (how to choose)

The number of ways to choose 3 noncollinear points out of 7 given points is: \(C_7^3 – C_4^3 – C_3^3 = 30\) (how to choose)

So the number of possible triangles is : 30 (triangles)

Solve the exercises Lesson 4, page 20, Math textbook 10 Kite episode 2

In the plane, let 6 parallel lines and 8 parallel lines are perpendicular to those 6 lines. How many rectangles are formed?

Solution method

Step 1: Calculate the number of ways to choose 2 parallel lines out of 6 parallel lines

Step 2: Calculate the number of ways to choose 2 parallel lines out of 8 parallel lines that are perpendicular to the original 6 parallel lines

Step 3: Apply the multiplication rule

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Solution guide

Number of ways to choose 2 parallel lines out of 6 parallel lines is: \(C_6^2\) (how to choose)

Number of ways to choose 2 parallel lines out of 8 parallel lines that are perpendicular to the original 6 parallel lines is: \(C_8^2\) (how to choose)

Applying the multiplication rule, we have the number of rectangles that can be formed: \(C_8^2.C_6^2 = 420\) (rectangle)

Solve the exercise Exercise 5 page 20 Math textbook 10 Kite episode 2

Expand the following expressions:

a) \({\left( {4y – 1} \right)^4}\)

b) \({\left( {3x + 4y} \right)^5}\)

Solution method

a) Use Newton’s binomial expansion with \(n = 4\): \({\left( {a + b} \right)^4} = {a^4} + 4{a^3}b + 6{a^2}{b^2} + 4a{b^3} + {b^4}\)

b) Use Newton’s binomial expansion with \(n = 5\):\({\left( {a + b} \right)^5} = {a^5} + 5{a^4}b + 10{a^3}{b^2} + 10{a^2}{b^3} + 5a{b^4} + {b^5}\)

Solution guide

a) \({\left( {4y – 1} \right)^4} = {\left[ {4y + \left( { – 1} \right)} \right]^4} = 256{y^4} – 256{y^3} + 96{y^2} – 16y + 1\)

b) \({\left( {3x + 4y} \right)^5} = 243{x^5} + 1620{x^4}y + 4320{x^3}{y^2} + 5760{x ^2}{y^3} + 3840x{y^4} + 1024{y^5}\)

Solving exercises Lesson 6 page 20 Math textbook 10 Kite episode 2

A computer password is a sequence of characters (in order from left to right) chosen from: 10 digits, 26 lowercase letters, 26 uppercase letters, and 10 special characters. Ngan wants to create a computer password with a length of 8 characters including: the first 4 characters are 4 different digits, the next 2 characters are lowercase letters, the next 1 character. Another is a capital letter, the last character is a special character. How many ways do you have to set up a computer password?

Solution method

Step 1: Select the first 4 characters as 4 different digits from 10 digits (with sorted)

Step 2: Select next 2 characters from 26 lowercase letters

Step 3: Choose the next 1 character from 26 uppercase letters

Step 4: Choose the last 1 character from 10 special characters

Step 5: Apply the multiplication rule

Solution guide

+) The number of ways to choose the first 4 characters is: \(A_{10}^4\) (how to choose)

+) The number of ways to choose the next 2 characters is: \(C_{26}^1.C_{26}^1\) (how to choose)

+) The number of ways to choose the next 1 character is: \(C_{26}^1\) (how to choose)

+) The number of ways to choose the last 1 character is: \(C_{10}^1\) (how to choose)

+) Applying the multiplication rule, we have the number of possible passwords:

\(A_{10}^4.C_{26}^1.C_{26}^1.C_{26}^1.C_{10}^1\) (password)

Solving exercises Exercise 7 page 20 Math textbook 10 Kite episode 2

A high school organized a relay race between classes with the content of 4 x 100 m and required each team to consist of 2 boys and 2 girls. An was assigned by the teacher to select 4 students and arrange their running order to register for the contest. How many ways do you An create a qualifying competition team? Know that An’s class has 22 boys and 17 girls.

Solution method

Step 1: Choose any 2 boys from 22 boys

Step 2: Choose any 2 girls from 17 girls

Step 3: Arrange the 4 you have chosen in some order

Step 4: Apply the multiplication rule

Solution guide

+) Number of ways to choose any 2 boys from 22 boys is: \(C_{22}^2\) (how to choose)

+) Number of ways to choose any 2 girls from 17 girls is: \(C_{17}^2\) (how to choose)

+) The number of ways to arrange the competition order of 4 friends is: \(4!\) (sorting)

+) Applying the multiplication rule, we have the number of ways to form a team: \(C_{22}^2.C_{17}^2.4!\) (how to make )

Solve exercises Exercise 8 page 20 Math textbook 10 Kite episode 2

Uncle Thao wants to buy 2 computers for work. The salesman introduced you to 3 computer manufacturers for your reference: the first company has 4 suitable computers, the second company has 5 suitable computers, the third company has 7 types of suitable computers. How many ways do you have to choose 2 computers for work?

Solution method

Choose any 2 computers from 4+5+7=16 computers => convolution 2 of 16.

Solution guide

+) Total number of matching computers is : \(4 + 5 + 7 = 16\) (computers)

+) The number of ways to choose 2 computers from 16 matching computers is: \(C_{16}^2 = 120\) (how to choose)

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