# Binomial Theorem Notes for IIT JEE, Download PDF!

JEE Main Short Notes

**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 }= ^{n}C_{0 }x^{n-0} y^{0} + ^{n}C_{1 }x^{n-1} y^{1} + ..... +^{n}C_{r }x^{n-r} y^{r} + ......^{n}C_{n }x^{0} y^{r} , i.e (x+y)^{n }= ∑ ^{n}C_{0 }x^{n-r} y^{r}

General term is T_{r+1} = ^{n}C_{r }x^{n-r} y^{r }Here ^{n}C_{0 ,} ^{n}C_{1 ,}^{n}C_{r , }^{n}C_{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.

^{n}C_{r} = ^{n}C_{(n-}_{r)}

^{n}C_{r} + ^{n}C_{(n-}_{r)} = ^{n+1}C_{r}

^{n}C_{r} / ^{n}C_{r-1} = (n - r + 1) / r

r (^{n}C_{r}) = n ( ^{n-1}C_{r-1} )

^{n}C_{r} / (r+1) = ( ^{n+1}C_{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)} = ^{n}C_{(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 )^{th}^{h} terms are both middle terms

They are given by

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

T_{((n + 3) / 2)} = ^{n}C_{((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 = ^{n}C_{(n/2)}

(b) If n is odd, then greatest coefficients are and ^{n}C_{((n-1)/2)} and ^{n}C_{((n+1)/2)}

**4.Binomial Coefficients**

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

Since (1+x)^{n} = ^{n}C_{0 } + ^{n}C_{1 }x + ^{n}C_{2 }x^{2} + ..... +^{n}C_{n }x^{n}

Put x=1

^{n}C_{0 } + ^{n}C_{1 }+ ^{n}C_{2 } + ..... +^{n}C_{n }=2^{n}

Put x=-1

^{n}C_{0 } + ^{n}C_{2 } + ^{n}C_{4 }+.....=^{n}C_{1 }+ ^{n}C_{3 }+ ^{n}C_{5}

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

**5. Series Summation**

^{n}C_{0 } + ^{n}C_{1 }+ ^{n}C_{2 } + ..... +^{n}C_{n }=2^{n}

**(b) **^{n}C_{0} - ^{n}C_{1} +^{n}C_{2} - ^{n}C_{3} + ... + (-1)^{n} ^{n}C_{n} = 0

**(c)** ^{n}C_{0 } + ^{n}C_{2 } + ^{n}C_{4 }+.....=^{n}C_{1 }+ ^{n}C_{3 }+ ^{n}C_{5 }= 2^{n-1}

**(d) **^{n}C^{2}_{0 } + ^{n}C^{2}_{1 }+ ^{n}C^{2}_{2 } + ..... +^{n}C^{2}_{n }=(2n) ! /(n!)^{3}

**6. Multinomial Expansion**

If n is a positive integer a_{1} , a_{2} ,... a_{m} are real or imaginary then

(a_{1} + a_{2} + a_{3} +....+a_{m} )^{n} = ∑(n! / n_{1}! n_{2}! n_{3}!...n_{m}! ) a_{1}^{n1 }a1^{n1 }...... a_{1}^{n1}

Where n_{1} , n_{2 }, n_{3 },...n_{m} are non-negative integers such that n_{1 }+ n_{2 }+ n_{3 }+ .... + n_{m} = n

(a) The coefficients of a_{1}^{n1 }a_{2}^{n2 }...... a_{m}^{n1 }in the expansion of (a_{1} + a_{2} + a_{3} +....+a_{m} )^{n} is (n! / n_{1}! n_{2}! n_{3}!...n_{m}! )

(b) The number of dissimilar terms in the expansion of (a_{1} + a_{2} + a_{3} +....+a_{m} )^{n} = ^{n+m-1}C_{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! ) x^{2} + ( (n(n-1)(n-2)) / 3! ) x^{3} + .... + ( (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 + x^{2} - x^{3} + x^{4} - .......

**(b) **(1 - x)^{-1} = 1 + x + x^{2} + x^{3} + x^{4} - .......

**(c) **(1+x)^{-2} = 1 - 2x + 3x^{2} - 4x^{3} + 5x^{4} - .......

**(d) **(1 - x)^{-1} = 1 + 2x + 3x^{2} + 4x^{3} + 5x^{4} - .......

**(e) **(1 + x)^{(1/2)} = 1 + (1/2)x + ( (1/2)(-1/2) / 2! )x^{2} + ( (1/2)(-1/2)(-3/2) / 3! )x^{3} +......

**(f)** (1 + x)^{(-1/2)} = 1 - (1/2)x + ( (1/2)(3/2) / 2! )x^{2} + ( (1/2)(3/2)(-3/2) / 3! )x^{3} +......

**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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Neha Bagri