🖋️MTS101. BINOMIAL THEOREM

-

 *BINOMIAL THEOREM* 

There are some subtopics which will make us better understand this topic. And they are:-

➕ FACTORIAL 

➖ PERMUTATION

➗ PASCAL TRIANGLE

✖️ BINOMIAL THEOREM

🟰 COMBINATION

🕺 Binomial Theorem itself

 _Lets treat them one after the other_ 

-----------------------------------------➕

 *FACTORIAL* 

Its denoted by !

And It is the multiple of terms from a given term to the lowest counting number

E.g 3!= 3×2×1

But note: 0! = 1! =1📌

 *Example*

1. Calculate 5! / 2! 

Ans:-

 5*4*3*2*1 / 2*1  

Then you'll be left with 5*4*3 

= *60* ✅


2.Calculate 10! × 3! / 8!   

Show workings

Try that

-----------------------------------------

Since we now understand factorial better, this will then lead us to Permutation

-----------------------------------------

➖ *PERMUTATION* 

*PERMUTATION* 

 _General formula_ 

log ) ⁿPr =  n! / (n-r)!

 *Note:* a. n is always greater than r📌

b. Both can't be negative


 *Examples*

1. Calculate ⁵P₂? 

Ans:- 

5! / (5-2)!

= 5! /3!

= *20* ✅


2. Solve ⁹P ₁₀?

 *Incorrect*✅ because r can't be greater than n


3. Solve ⁵P₃₋ₓ = 120

Ans:-  

5! / (3-x)! = 120

So since 5! = 120

That's 

120/ (5-(3-x))! = 120 

using arithmetics

You'll arrive at :

1/ (2+x)! = 1

Therefore

(2+x)! = 1

Since 1! = 1

So (2+x)! = 1!

Factorial will then cancel

2+x = 1

 *x = -1✅*

Now we understand this

++++++++++++++++++++++

➗ *PASCAL TRIANGLE*

There is a diagram for this. Observe it and understand it.

Let's treat some examples on it.

Note: you can only use pascal Triangle if the power is not more than 7.📌


 *Example* 

1. *Expand (x + 2)³ using Pascal Triangle*

Ans:

= x³2⁰ + x²2¹ + x¹2² + x⁰2³

= x³ + 2x² + 4x¹ + 8

This isn't all cos we're using Pascal Triangle

Therefore it'll be coefficient of power 3 since the question is (x +2)³

= 1,3,3,1

Using pascal Triangle

So you'll the add to this 

= x³ + 2x² + 4x¹ + 8

= 1x³ + 3(2x²) + 3(4x¹) + 8

 *= x³ + 6x² + 12x¹ + 8✅*

I guess this is explanatory

2. *Expand (x - 2)³ using Pascal Triangle*

Ans:

You'll take same process as first but🤔 instead of using directly, it'll be (x + (-2))³

= x³(-2)⁰ + x²(-2)¹ + x¹(-2)² + x⁰(-2)³

= x³ - 2x² + 4x¹ - 8

Coefficients of power 3

= 1x³ - 3(2x²) + 3(4x¹) - 8

 *= x³ - 6x² + 12x¹ - 8✅*


 *EXERCISE*

Solve the following using pascal Triangle

1.  (2x + y)⁴

2.  (3 - x)³

3. (x + y)⁶ 


Guys guess what😂

We've started the next subtopic.

Pascal Triangle method is still same as Binomial Expansion

++++++++++++++++++++

✖️ *BINOMIAL EXPANSION*

Like I said it's same as Pascal Triangle✍️ 


××××××××××××××××××××××

🟰 *COMBINATION*

Permutation method already opened our eyes to this.


 _General formula_ 

ⁿCr = n! / (n-r)! r!


 *Examples*

1. 5C3 ?

= 5!/ (5-3)! 3!

=5!/ 2! 3!

= 5×4/2

= *10* ✅ 


2. 5C7 ?

Ans: *incorrect*✅ because r can't be greater than n


3. *Expand (x + 2)³ using combination method*

Solution:- 

 ³C₀x³2⁰ +  ³C₁x²2¹ + ³C₂x¹2² +  ³C₃x⁰2³


Finish it up 


4. Using Combination method

Expand (2x + y)⁴


 _To be continue_

Written by:

          Mr. Peter Olabiyi


✍️ *FLASHPEE EDUCATIONAL TEAM*

Comments

Post a Comment

Popular posts from this blog

📝 GNS101

-
21ST CENTURY LITERACY SKILLS  There are 4 language skills:  1. Listening 2. Speaking 3.Reading 4. Writing.  *The four skills are related to each other by two parameters:*  (a). the mode of communication: oral or written   (b) the direction of communication: receiving or producing the  message   *PRODUCTIVE SKILLS*  The productive skills are  1. SPEAKING 2. WRITING Learners doing these need to produce language.  These skills are contrasted with the *receptive skills* of listening and reading.    *RECEPTIVE SKILLS*  The receptive skills are  1. LISTENING 2. READING Learners do not need to produce language to do these: they receive and understand it.  They can be contrasted with the productive of speaking and writing.  These skills are sometimes erroneously described as   *passive skills,* as if they require no activity.  Note that all skills require some activity.   1. *LITERACY*  ...
Read more

📓 GNS101 WORD FORMATION (Morpheme)

-
WORD FORMATION (Morpheme)  *Word formation* refers to the way in which new words are formed in the basis of other words or morpheme.  And a morpheme is meaningful linguistic unit consisting of a word that cannot be divided into smaller meaning parts  _There is a difference between a morpheme and a word_  ▪️ *A word* can stand alone and can be meaningful but *a morpheme* can not always stand alone ▪️ *A word* can have 2 or more morphemes   ```LET ME GIVE EXAMPLE```   *RECONSTRUCT*    This is a word that has two morphemes (RE/CONSTRUCT).   When we have a sentence, it consists of clause. ⚡Clause consists of phrase ⚡Phrases consists of words ⚡And words consist of morpheme ⚡Morpheme consists of syllables   *Define a phrase*    Group of words without finite verb. So we can break this to phrase They/ might be thinking/of words/ and their meaning   Each of these phrases consists of one or two words   So words can be broken to...
Read more

📝GNS101. GOAL SETTING

-
GOAL SETTING AND TIME MANAGEMENT  *What is Time?* *Time* is a *limited* , *non-renewable* resource.  ❗❗❗. Note:  those key words must be there *What is Management?*  *Management* is the skilful handling or use of resources such as people (human capital), money,  materials and time  *What is Time Management?* *Time management* is an essential skill in the modern world where several activities have to be crammed into each day.  *There are several time management tools you can use.*  They are cheap and easily available. 🔘 a. *TIMETABLE* :  this is a tabulated list of the times at which certain events or activities are required to occur.  🔘b. *ORGANIZER* : This is a small portable calendar and data book used for planning, or a handheld  computerized device with a simple database for managing appointments and other information.  🔘 c. *TIMEPIECE* : A timepiece is any instrument for recording or showing the time, especially one tha...
Read more