Binomial theorem

Before we start this blog, lets dig deep ( and If you were part of the covid batch very deep) into our highschool maths memory. Ofcourse , the memory of your maths teacher yelling at everyone for making silly mistakes like 3*3 = 6 or Your mind wondering why trigonometric equations have more alphabets than numbers might be the first to resurface. Though they are very nostagic memories, I would request you to push them aside for a little while and focus on the green board behind the yelling teacher.

a plus b whole square is equal to a square + b square + two a b. Remember the monotone song you used to song while telling that formula? Yup, hold onto that memory...that's going to be the star of this blog.

We know that

And for them who were really into maths and went one step ahead, might also remember this :-

Now, what If I ask you what is (a+b) ^ 4 ? That's easy right, Just multiply (a+b) ^ 3 with (a+b) and voila! Done

Hmm.. I am not going to let you go that easy, how about (a+b) ^ 18?

Stumped right? But what if I tell you there is an easier way of solving the problem rather than multiplying the number 18 times ( or wishing a meteor falls on your school so that you don't have to expand this in your exam)

Welcome to the world of Binomial Theorem!

And then the security in front of this world stops you and asks "Tell me the secret code of ncr" (and you turn to me, ready to punch, hold on! I can explain!)

The NCR rule

The nCr represents the selection of objects from a group of objects where order of objects does not matter. So for example, if there are 4 different object and at a time a person can take two of these objects, How many combinations are possible? By counting, we would find 6 pairs are possible. Now as the total number of things increase, so does the number of possibilities ( and our headache while counting each one)

Hence, there is a simple formula we can use to find it out,

Here n represents the total number of items and r represents that number chosen at a time

( ! tells us to take the factorial , for example 5 ! = 5 * 4* 3* 2* 1 . We should take the product of all the natural numbers less than and equal to the given number)

We can use this formula in the example we used before, lets take n =4 and r = 2

which after calculations yield 6.

You might be wondering how this is anywhere related to the Binomial theorem?

This NCR rule will help us figure out the coefficients of each term in the expansion.

(Coeffecient = It is the constant or numerical term multiplied with the algebraic term . Egs:- 4x , Coefficient = 4)

To understand this let's take the coefficients of the expression (a+b)^0 = 1 ( or 1x^0) , (a+b) , (a+b) ^2 and (a +b ) ^3

The coefficients will be

(a+b)^0 = 1 ( Coefficient = 1)

(a+b) = 1 and 1

(a+b) ^2 = 1 , 2 and 1

(a +b ) ^3 = 1 ,3 , 3 and 1

Which can also be written in this form,

Known as the Pascal triangle,

These coefficients can also be derived using the NCR rule

For better visualisation, let's take an example

we all know the expansion of (a+b) ^3 by now... but this can also be written as:-

When we solve the coefficients with the NCR formula , we will get 1, 3 ,3 and 1

Notice any pattern?

* The n in the formula remains constant throughout and is the power with which the original expression is raised.

* The r increases from 0 to the value of n

* The power of a decreases by 1

* The power of b increases by 1 throughout

By following this pattern, we can find the expansion of a binomial expression raised to any power!

Now that we have even made numbers exclaim, You might wonder how this is useful in real life?

*Statistics and Probability heavily depend on this theorem

*Architecture

*Weather forecasting and etc

Phew, I guess that's the end of my thinking capacity for now... I hope this binomial theorem helps increase your maths average exponentially! （￣︶￣）↗