Binomial Theorem

Binomial Theorem

​Introduction

​Bino=2

Pascal's Triangle!

The secret to expanding binomials!

​Notes

• Index = n

Total no of terms = n+1

• From left to right, a↓, b ↑

​Pascal's Triangle

​Pascal's Triangle

​Continuation...

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Binomial Theorem:

• The binomial theorem is used to expand expressions of the form (a + b)^n.
• It allows us to find the coefficients of each term in the expansion.
• The expanded form of the expression a + 4b^3 using the binomial theorem is a^3 + 12a^2b + 48ab^2 + 64b^3a + b^3.

Mastering the Binomial Theorem

To find the 5th coefficient of the expansion form of 2𝑎+3𝑏20, we use the formula 𝑛𝑘−1𝑎𝑛−(𝑘−1)𝑏(𝑘−1). In this case, 𝑛=20 and 𝑘=5. Plugging these values into the formula, we get 20𝑎15𝑏4. Therefore, the 5th coefficient is 20.

Multiple Choice

What is the 5th coefficient of the expansion form of 2𝑎+3𝑏20?

• 20𝑎15𝑏4

• 20𝑎14𝑏5

• 20𝑎16𝑏3

• 20𝑎13𝑏6

5th Coefficient:

Trivia: The 5th coefficient of the expansion form of 2𝑎+3𝑏20 is 20𝑎15𝑏4. This means that when the expression is expanded, the term with the 5th coefficient will have 20𝑎 raised to the power of 15 and 3𝑏 raised to the power of 4. It's interesting how the coefficients and exponents combine to form the expanded expression!

Mastering the Binomial Theorem

• Binomial Theorem: 𝑎+𝑏𝑛=𝑘=0𝑛𝑛𝑘𝑎𝑛−𝑘𝑏𝑘
• Example: 𝑎+𝑏3=𝑎3+3𝑎2𝑏+3𝑎𝑏2+𝑏3
• Expand: 𝑎+4𝑏3 using binomial theorem

Multiple Choice

What is the expansion of 𝑎+4𝑏³ using the binomial theorem?

• 𝑎⁴+4𝑎³𝑏+6𝑎²𝑏²+4𝑎𝑏³+𝑏⁴

• 𝑎⁴+4𝑎³𝑏+6𝑎²𝑏²+4𝑎𝑏³

• 𝑎⁴+4𝑎³𝑏+6𝑎²𝑏²+4𝑎𝑏³+𝑏⁴+4𝑏⁴

• 𝑎⁴+4𝑎³𝑏+6𝑎²𝑏²+4𝑎𝑏³+𝑏⁴+4𝑏⁴+4𝑏⁴

Binomial Theorem:

• The expansion of 𝑎+4𝑏³ using the binomial theorem is:
• 𝑎⁴+4𝑎³𝑏+6𝑎²𝑏²+4𝑎𝑏³+𝑏⁴
• This theorem helps in expanding expressions with binomial coefficients.
• It is widely used in algebra and calculus.

Mastering the Binomial Theorem

• Expand the expression 𝑎+4𝑏3 using binomial theorem
• 𝑎+4𝑏3 = 𝑎3+12𝑎2𝑏+48𝑎𝑏2+64𝑏3
• 𝑎+𝑏3 = 30𝑎3−0𝑏0+31𝑎3−1𝑏1+32𝑎3−2𝑏2+33𝑎

Multiple Choice

What is the expanded form of the expression 𝑎+4𝑏3 using the binomial theorem?

• 𝑎3+12𝑎2𝑏+48𝑎𝑏2+64𝑏3𝑎+𝑏3 = 30𝑎3−0𝑏0+31𝑎3−1𝑏1+32𝑎3−2𝑏2+33𝑎

• 𝑎3+12𝑎2𝑏+48𝑎𝑏2+64𝑏3𝑎+𝑏3 = 30𝑎3−0𝑏0+31𝑎3−1𝑏1+32𝑎3−2𝑏2+33𝑎

• 𝑎3+12𝑎2𝑏+48𝑎𝑏2+64𝑏3𝑎+𝑏3 = 30𝑎3−0𝑏0+31𝑎3−1𝑏1+32𝑎3−2𝑏2+33𝑎

• 𝑎3+12𝑎2𝑏+48𝑎𝑏2+64𝑏3𝑎+𝑏3 = 30𝑎3−0𝑏0+31𝑎3−1𝑏1+32𝑎3−2𝑏2+33𝑎

Binomial Theorem:

• The binomial theorem is used to expand expressions of the form (a + b)^n.
• It allows us to find the coefficients of each term in the expansion.
• The expanded form of the expression a + 4b^3 using the binomial theorem is a^3 + 12a^2b + 48ab^2 + 64b^3a + b^3.

Mastering the Binomial Theorem

To find the 5th coefficient of the expansion form of 2𝑎+3𝑏20, we use the formula 𝑛𝑘−1𝑎𝑛−(𝑘−1)𝑏(𝑘−1). In this case, 𝑛=20 and 𝑘=5. Plugging these values into the formula, we get 20𝑎15𝑏4. Therefore, the 5th coefficient is 20.

Multiple Choice

What is the 5th coefficient of the expansion form of 2𝑎+3𝑏20?

• 20𝑎15𝑏4

• 20𝑎14𝑏5

• 20𝑎16𝑏3

• 20𝑎13𝑏6

5th Coefficient:

Trivia: The 5th coefficient of the expansion form of 2𝑎+3𝑏20 is 20𝑎15𝑏4. This means that when the expression is expanded, the term with the 5th coefficient will have 20𝑎 raised to the power of 15 and 3𝑏 raised to the power of 4. It's interesting how the coefficients and exponents combine to form the expanded expression!

Multiple Choice

Expand the expression using the Binomial Theorem.
(2x+5)⁴

• 2x⁴ + 40x³ + 300x² + 1000x +625

• 16x⁴ + 160x³ +600x² +1000x + 625

• 16x⁴ + 1000x³ + 600x² + 160x +625

Combination refers to a list of numbers where the order doesn't matter at all.

Multiple Choice

Expand the expression using the Binomial Theorem.

(2x+5)⁴

• 2x⁴ + 40x³ + 300x² + 1000x +625

• 16x⁴ + 160x³ +600x² +1000x + 625

• 16x⁴ + 1000x³ + 600x² + 160x +625

Multiple Choice

Expand the expression using the Binomial Theorem.
(2x+5)⁴

• 2x⁴ + 40x³ + 300x² + 1000x +625
• 16x⁴ + 160x³ +600x² +1000x + 625
• 16x⁴ + 1000x³ + 600x² + 160x +625

Multiple Choice

What is row 7 of Pascal's Triangle?

• 1, 5, 10, 5, 1
• 1, 5, 10, 10, 5, 1
• 1, 7, 21, 35, 35, 21, 7, 1
• 1, 7, 21, 35, 21, 7, 1

Multiple Choice

Expand: $$\left(a+b\right)^2$$  

• $$a^2+ab+b^2$$  

• $$a^2+2ab+b^2$$  

• $$a^2-ab+b^2$$  

• $$a^2-2ab+b^2$$  

Multiple Choice

Choose the right Pascal's triangle

Multiple Choice

Expand: $$\left(a-b\right)^5$$  

• $$a^5+5a^4b+10a^3b^2+10a^2b^3+5ab^4+b^5$$  

• $$a^5-5a^4b+10a^3b^2-10a^2b^3+5ab^4-b^5$$  

• $$a^5-5a^4b+10a^2b^3-10a^3b^2+5ab^4-b^5$$  

• $$a^5-5a^4b-10a^3b^2-10a^2b^3-5ab^4-b^5$$  

Multiple Choice

Use the binomial theorem to expand $$\left(4-2z\right)^3$$  

• $$64-96z+48z^2+8z^3$$  

• $$64-96z+48z^2-8z^3$$  

• $$64+96z+48z^2+8z^3$$  

• $$64-64z+144z^2-8z^3$$  

Multiple Choice

Find the product:
(y - 3)(y + 7)

• y² - 21
• y² + 10x - 21
• y² + 4y - 21
• 2y - 10

Multiple Choice

Find the product:

(3x² – 1)(3x² + 5x+4)

• 6x² + 4x + 5

• 9x⁴ + 15x³ - 4x

• 6x² + 15x³ - 20x² - 5x + 4

• 9x⁴ + 15x³ - 9x² - 5x - 4

Multiple Choice

What would be the 5th term of the expansion of (n+4)⁴?

• 257

• 256

• n⁴

• 259

Multiple Choice

What would be the 2nd term of the expansion of (y+4)⁴?

• 253y

• 16y³

• 64y

• 96y²

Multiple Choice

What would be the 4th term of the expansion of (a+3)⁴?

• 108a

• 12a³

• 81

• 27a

Multiple Choice

Number of Combination

1. A person is going to a candy shop where there are 7 types of flavors, if this person is only going to buy 3, define every combination possible.

• C = 35

• C = 84

Multiple Choice

Number of Combination

2. Two girls will go to a party, if between the two, they have 4 pairs of fancy shoes, define the combination of shoes this two girls can wear

• C = 6

• C = 10

Multiple Choice

Number of Combination

3. A man will go on a trip for 3 days, so he will take with him 3 shirts, if he has 7 shirts, how many combination of shirts can he take?

• C = 35

• C = 84

Multiple Choice

Number of Combination

5. A sportsman goes to the store to buy 4 pairs of shoes, if at the store there are a lot of shoes in 7 available colors, how many combination of colors can this man buy.

• C = 35

• C = 210

Multiple Choice

Students falling in line during the flag ceremony

• Combination

• Permutation

Multiple Choice

Batting order in a baseball game

• Permutation

• Combination

Multiple Choice

Evaluate 10C8

• 10

• 45

• 90

• 1 814 400

Multiple Choice

Rhina's mom gave her seven keychains bought from Baguio. If she decided to put five of them in her backpack, in how many ways can she do it?

• 21

• 42

• 1260

• 2520

Multiple Choice

If Marie has 10 pre-loved clothes how many ways can she choose five of them to give to her younger cousin.

• 25

• 125

• 252

• 30 240

Multiple Choice

You need to bring 4 from 8 different shoes in your friends pageant. How many possible combination of shoes can you bring?

• 16

• 56

• 70

• 1680

Multiple Choice

Evaluate: 6C2

• 15

• 30

• 360

• 720

Multiple Choice

Evaluate: 3C1

• 1

• 2

• 3

• 4

Multiple Choice

What is the formula for Combination?

• n!

• n!/(n-r)!

• n!/(n-r)!r!

• n!/r!(n-r)!r!

Multiple Choice

Which of the following situation involves combination?

• Arranging chairs in a circular table

• Friends sitting on the front row of movie theater

• Choosing three students in class committee

• Creating a 4-digit gcash MPIN passcode

Multiple Choice

It is the number of ways of selecting from a set when the order

is not important

• Combination

• Fundamental Counting Principle

• Permutation

• Probability

Multiple Choice

 A group of 3 lawn tennis players S, T, U. A team consisting of 2 players is to be formed. In how many ways can we do so?

• 3

• 20

• 5

• 6

Multiple Choice

Find the number of subsets of the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} having 3 elements.

• 720

• 120

• 100

• 60

Multiple Choice

uppose we have a set of 6 letters { A,B,C,D,E,F}. In how many ways can we select a group of 3 letters from this set?

• 3

• 6

• 30

• 60