Binomial Theorem Notes for IIT JEE, Download PDF!

Binomial theorem is an important and basic formula in algebra. It has its applications in other topics as well. Thus, it is very important for a JEE Main aspirant to prepare this topic in a well-versed manner. These notes are also useful in your JEE Advanced and BITSAT preparation. You can also download Binomial Theorem Notes PDF for IIT JEE preparation at the end of the post.

1. Binomial Theorem for positive integral index

If n is a positive integer and x, y is real or imaginary then (x+y)ⁿ = ⁿC₀ x^n-0 y⁰ + ⁿC₁ x^n-1 y¹ + ..... +ⁿC_r x^n-r y^r + ......ⁿC_n x⁰ y^r , i.e (x+y)ⁿ = ∑ ⁿC₀ x^n-r y^r

General term is T_r+1 = ⁿC_r x^n-r y^r Here ⁿC_{0 ,} ⁿC_{1 ,}ⁿC_{r ,} ⁿC_n are called the binomial coefficients.

Replacing y by -y the general term of (x-y)ⁿ is obtained as

T_r+1 = (-1)^{r n}C_r x^n-r y^r

Similarly, the general term of (1 + x)ⁿ and (1 - x)ⁿ can be obtained by replacing x by 1 and x by 1 and y by -x respectively.

ⁿC_r = ⁿC_(n-r)

ⁿC_r + ⁿC_(n-r) = ⁿ⁺¹C_r

ⁿC_r / ⁿC_r-1 = (n - r + 1) / r

r (ⁿC_r) = n ( ^n-1C_r-1 )

ⁿC_r / (r+1) = ( ⁿ⁺¹C_r+1 ) / (r+1)

2. Middle Term

The middle term depends upon the even or odd nature of n

Case 1: When n is even

Total number of terms in the expansion of (x+y)ⁿ is n+1 (odd)

So there is only 1 middle term i.e ((n/2) + 1)^th term is the middle term

This is given by

T_{((n/2) + 1)} = ⁿC_(n/2) x^(n/2) y^(n/2)

Case 2: When n is odd

Total number of terms in the expansion of (x+y)ⁿ is n+1(even)

So there are 2 middle terms i.e ((n + 1) / 2 )^th and((n + 3) / 2 )^thh terms are both middle terms

They are given by

T_{((n + 1) / 2)} = ⁿC_{((n-1) / 2)} x^{((n+1) / 2)} y^{((n-1) / 2)}

T_{((n + 3) / 2)} = ⁿC_{((n+1) / 2)} x^{((n-1) / 2)} y^{((n+1) / 2)}

3. Greatest Term

If T_r and T_r+1 be the r^th and (r+1)^th terms in the expansion, (x+y)ⁿ then

T_r+1 / T_r = (( n - r + 1 ) / r ) x

Let T_r+1 be the numerically greatest term in the above expression

Then

T_r+1≥T_r

Or

(( n - r + 1 ) / r ) x

Greatest Coefficient:

(a) If n is even, the greatest coefficient = ⁿC_(n/2)

(b) If n is odd, then greatest coefficients are and ⁿC_((n-1)/2) and ⁿC_((n+1)/2)

4.Binomial Coefficients

In the binomial expansion of (1 + x)ⁿ let us denote the coefficients by ⁿC_{0 ,} ⁿC_{1 ,}ⁿC_{r ,} ⁿC_n respectively

Since (1+x)ⁿ = ⁿC₀ + ⁿC₁ x + ⁿC₂ x² + ..... +ⁿC_n xⁿ

Put x=1

ⁿC₀ + ⁿC₁ + ⁿC₂ + ..... +ⁿC_n =2ⁿ

Put x=-1

ⁿC₀ + ⁿC₂ + ⁿC₄ +.....=ⁿC₁ + ⁿC₃ + ⁿC₅

Sum of odd terms coefficients=Sum of even terms coefficients=2^n-1

5. Series Summation

ⁿC₀ + ⁿC₁ + ⁿC₂ + ..... +ⁿC_n =2ⁿ

(b) ⁿC₀ - ⁿC₁ +ⁿC₂ - ⁿC₃ + ... + (-1)^{n n}C_n = 0

(c) ⁿC₀ + ⁿC₂ + ⁿC₄ +.....=ⁿC₁ + ⁿC₃ + ⁿC₅ = 2^n-1

(d) ⁿC²₀ + ⁿC²₁ + ⁿC²₂ + ..... +ⁿC²_n =(2n) ! /(n!)³

6. Multinomial Expansion

If n is a positive integer a₁ , a₂ ,... a_m are real or imaginary then

(a₁ + a₂ + a₃ +....+a_m )ⁿ = ∑(n! / n₁! n₂! n₃!...n_m! ) a₁^n₁ a1ⁿ¹ ...... a₁^n₁

Where n₁ , n₂ , n₃ ,...n_m are non-negative integers such that n₁ + n₂ + n₃ + .... + n_m = n

(a) The coefficients of a₁^n₁ a₂^n₂ ...... a_m^n₁ in the expansion of (a₁ + a₂ + a₃ +....+a_m )ⁿ is (n! / n₁! n₂! n₃!...n_m! )

(b) The number of dissimilar terms in the expansion of (a₁ + a₂ + a₃ +....+a_m )ⁿ = ^n+m-1C_m-1

7. Binomial theorem for negative or fractional index

When n is negative and/or fraction and |x|<1 then

(1+x)ⁿ = 1 + nx + ( (n(n-1) ) / 2! ) x² + ( (n(n-1)(n-2)) / 3! ) x³ + .... + ( (n(n-1)(n-2).....(n-r+1)) / r! ) x^r + ...........upto infinity

The general term is

T_r+1 = ( (n(n-1)(n-2).....(n-r+1)) / r! ) x^r

For example, if |x|<1 then,

(a) (1+x)^-1 = 1 - x + x² - x³ + x⁴ - .......

(b) (1 - x)^-1 = 1 + x + x² + x³ + x⁴ - .......

(c) (1+x)^-2 = 1 - 2x + 3x² - 4x³ + 5x⁴ - .......

(d) (1 - x)^-1 = 1 + 2x + 3x² + 4x³ + 5x⁴ - .......

(e) (1 + x)^(1/2) = 1 + (1/2)x + ( (1/2)(-1/2) / 2! )x² + ( (1/2)(-1/2)(-3/2) / 3! )x³ +......

(f) (1 + x)^(-1/2) = 1 - (1/2)x + ( (1/2)(3/2) / 2! )x² + ( (1/2)(3/2)(-3/2) / 3! )x³ +......

8. Using Differentiation and Integration in Binomial Theorem

(a) Whenever the numerical occur as a product of binomial coefficients, differentiation is useful.

(b) Whenever the numerical occur as a fraction of binomial coefficients, integration is useful

Binomial Theorem Notes for JEE Main Download PDF

Moreover, you can also check the syllabus for JEE Main exam through the links shared below:

JEE Main Syllabus with weightage

JEE Main Question Paper 2019 with Solutions

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