JEE Maths Binomial Theorem and its Simple Applications
Bionomial Theorem and its Simple Applications PDF Notes, Important Questions and Synopsis
SYNOPSIS
 A binomial is a polynomial having only two terms. For e.g 2y^{2} 1
 (x + y)^{n} can be expanded using the Binomial theorem without actually multiplying it n times.

Properties of Binomial Expansion (x + y)^{n}
i.Total number of terms in this expansion is n + 1.
ii.The exponent of x decreases by 1, while the exponent of y increases by 1 in subsequent terms.
iii.The first term is and the final/last term is
iv.The general term in this expansion is given by

Binomial Coefficients:
Binomial coefficients in the expansion of (x + y)^{n} are simply the number of ways of choosing x from the brackets and y from the rest. 
Pascal’s Triangle:
Binomial coefficients can be found using Pascal’s triangle given below.
We can also say that
.
Also, the binomial coefficient is given by

Terms in the binomial expansion of (x + y)n
i.General Term (r^{th} term): General term in the expansion of (x + y)^{n} is given byii.Middle Term(s):
1. When n is even,
2. When n is odd,
and
iii.Greatest Term:
In any binomial expansion, the values of the terms increase, reach a maximum and then decrease.
So, to find the greatest term, find the value of r till
So, the greatest term occurs when
r =
iv.Term independent of x:The term independent of x is the term not containing x.
So, find the value of r such that the exponent of x is zero. 
Applications of Binomial Expansion:
i. We have a very important result
, n ≥ 1, n ∊ N
ii. Finding the remainder using Binomial Theorem:
To find the remainder when pn is divided by q, adjust the power of p to p^{m} which is very close to b, say with difference 1 and then divide by taking the remainder always positive.
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