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.
Related Chapters
 Sets, Relations and Functions
 Complex Numbers and Quadratic Equations
 Matrices and Determinants
 Permutations and Combinations
 Mathematical Induction
 Sequences and Series
 Limit, Continuity and Differentiability
 Integral Calculus
 Differential Equations
 Coordinate Geometry
 Three Dimensional Geometry
 Vector Algebra
 Statistics and Probability
 Trigonometry
 Mathematical Reasoning