Binomial Theorem | free study material for IIT JEE , NDA

Binomial Theorem

What is Binomial Expression ?
An algebraic expression consisting of two terms with +ve or -ve sign between them is called a binomial expression.
For example : ( a + b) , ( 2x -3y )
Binomial theorem for positive integral index :


Here ⁿC₀ , ⁿC₁ , ⁿC₂ ,........ⁿC_n are called binomial coefficients
General term from the begining of the binomial expansion : The general term of the expansion is (r+1)^th term usually denoted by
T_r+1 = ⁿC_ra^n-rb^r
General term from the end of the binomial expansion : The general r^th term from end of the expansion is usually denoted by

Middle term :
Middle term depends upon the value on n
1. When n is even : the total number of terms in the expansion of ( a + b )ⁿ is n + 1 (odd) , So there is only one middle term :

2. When n is odd : the total number of terms in the expansion of ( a + b )ⁿ is n + 1 (even) , So there is two middle term :

Total number of terms
For binomial expression (a₁ + a₂ + a₃ + ...... + a_k)ⁿ
Total number of terms = ^n+k-1C_k-1
Properties of coefficient
In the binomial expansion of ( 1 + x )ⁿ = ⁿC₀ + ⁿC₁x + ⁿC₂x² + ........ + ⁿC_nxⁿ
1. The sum of binomail coefficients in the expansion of (1 + x)ⁿ is 2ⁿ
Put x = 1 in (1 + x)ⁿ , we get 2ⁿ = ⁿC₀ + ⁿC₁ + ⁿC₂ + ........ + ⁿC_n
2ⁿ = ⁿC₀ + ⁿC₁ + ⁿC₂ + ........ + ⁿC_n 2. Sum of binomial coefficients with alternate sign :
putting x = -1
we get , ⁿC₀ - ⁿC₁ + ⁿC₂ - ⁿC₃ ........ = 0
ⁿC₀ - ⁿC₁ + ⁿC₂ - ⁿC₃ ........ = 0 3. Sum of the coefficients of the old terms in the expansion of (1 + x)ⁿ is equal to sum of the coefficients of even terms and each is equal to 2^n-1
nC₀ + ⁿC₂ + ⁿC₄ + ........ = 2^n-1
nC₁ + ⁿC₃ + ⁿC₅ + ........ = 2^n-1
nC₀ + ⁿC₂ + ⁿC₄ + ........ = 2^n-1
nC₁ + ⁿC₃ + ⁿC₅ + ........ = 2^n-1
Binomial expansion if n is not integer

a) (1+x)^-1 = 1 - x + x² - x³ + x⁴ -----------
b) (1+x)^-2 = 1 - 2x + 3x² - 4x³ -----------
c) (1-x)^-1 = 1 + x + x² + x³ + x⁴ -----------
d) (1-x)^-2 = 1 + 2x + 3x² + 4x³ -----------
Important Points :
1. The number of terms in the expansion of (a + b)ⁿ are n + 1 .
2. In any term of expansion of (a + b)ⁿ, the sum of exponents of a and b is always constant = n
3. The coefficients of x^n-1 in the expansion of (x-1)(x-2).....(x-n) = n(n+1)/2
Practice set 1 Practice set 2 Practice set 3