# CBSE Class 11 Physics Revision Notes Chapter 13

## Class 11 Mathematics Revision Notes for Chapter 8 Binomial Theorem

Mathematics is one of the most crucial subjects in Class 11. Students should study the subject properly to score good marks in the final papers and other entrance examinations. Amongst all the other chapters, the binomial theorem is considered one of the most scoring. So, candidates must practice all the numerals from the NCERT books. One of the best ways to strengthen one’s preparation is by checking the revision notes and regularly studying from them.

Class 11 Mathematics Revision Notes for Chapter-8 Binomial Theorem – Free PDF Download

## Access Class 11 Mathematics Chapter 8 – Binomial Theorem Notes

Binomial Theorem

An algebraic expression containing the two distinct terms connected to each other by an addition or subtraction operation is known as a binomial. The binomial theorem is the method of expanding an expression raised to any finite power. It is a powerful theorem when expansion becomes lengthy and difficult to calculate. With the binomial theorem, any expression that is raised to a very large power can be easily calculated. The theorem is very useful in Algebra, probability and other calculations.

In binomial expansion, we usually have to find the middle or general term. The terms that are used in the binomial expansion are:

• The ratio of consecutive terms/coefficients
• Numerically greatest term
• Particular term
• Determining a particular term
• Independent term
• Middle term
• General term

Binomial Theorem formula

The formula for the Binomial Theorem is:

(x + y)n = Σr=0n  nCr xn – r · yr.

### Binomial Theorem for Positive Integral Indices

The binomial theorem for the positive integral indices states that the total number of terms in the expansion is one more than the index.

The properties of the Binomial Theorem for positive integral indices are mentioned below.

They are as follows:

• (a + b)0 = 1                                       a +b is not equal to 0
• (a +b)1 = a + b
• (a + b)2 = a2 +2ab + b
• (a + b)3 = a3 + 3a2b + 3ab2 + b3
• (a + b)4 = (a + b)3 (a + b) = a4 + 4a3b + 6a2 b2 + 4ab3 + b

From the above, we observe that:

• The total terms present in the expansion are one more than the index.
• Powers of the first quantity, let’s say ‘a’ goes on reducing by 1 whereas the power of the second quantity ‘b’ will increase by 1 in the successive terms.
• In each term of the expansion the sum of indices, a and b will be the same and equal to the index of a + b.

### Pascal’s Triangle:

Pascal’s Triangle is an arrangement of the expansion coefficients in a triangular shape with (1) at the top of the vertex and running down the two slanting sides.

The expansions for higher powers of a binomial are possible through Pascal’s triangle.

### Binomial Theorem for any Positive Integer N

The binomial theorem for any positive integer n is

(a + b)n = n C0 an + n C1 an-1b + n C2 an-2 b2 + n Cn-1 a-b n-1 + nCn bn

### Properties of Binomial Coefficient:

Binomial coefficients are the integers that are coefficients in a binomial theorem. Some of the important properties of binomial coefficients are given below:

• C02 + C12 + C22 + …Cn2 = [(2n)!/ (n!)2]
• C1 − 2C2 + 3C3 − 4C4 + … +(−1)n-1 Cn = 0 for n > 1
• nC1 + 2.nC2 + 3.nC3 + … + n.nCn = n.2n-1
• C0 – C1 + C2 – C3 + … +(−1)n . nCn = 0
• C0 + C2 + C4 + … = C1 + C3 + C5 + … = 2n-1
• C0 + C1 + C2 + … + Cn = 2n

### Some Important Results:

Some important results derived are:

• (1 + X)n = nC0X0 + nC1X1 + nC2X2 + ……….. + nCnXn

Putting x =1 and -1 we will obtain

C0 + C1 + C2+ Cn = 2

C0 – C1 + C2 -……….. + (-1)n Cn = 0

• On differentiating students will get the following:

Differentiating the (1 + x)n = nC0Xo + nC1x1 + nC2X2 +…….. + nCnXn on both sides

n(1 + x)n-1

C1 + 2C1x+3C3X2 + ……. +nCNx n-1……. (1)

x=1

n2 n-1 = C1  + 2C1 + 3C3 + ……… + nCn

X = -1

0 = C1 – 2C1 + ……. + (-1) n-1 nCn

On differentiating 1 again and again we will get different results.

### Multinomial Expansion:

While expanding the (x1 + x2 + …….. + xn ) m  where m,n N  and 1 2 n x x … x    are independent variables.

• Total number of terms = m + n-1 C n-1
• Coefficient of X1 r X2 r2 ………. Xn rn

Where the r1 + r2 …… +rn  = m) is m, ri ∈ N U {0} is m!/ r1 ! r2 ! …….. Rn!

• The sum total of all the coefficients is obtained by putting the variables equals to 1.

### Binomial Expression

The binomial theorem is considered one of the greatest discoveries in the world of Mathematics. It was developed by Euclid in the 4th century. A Persian Mathematician Al-Karaji gave proof of the binomial theorems and Pascal’s triangle. Binomial expression is an expression consisting of two different terms. According to the binomial theorem for any particular positive integer n, the total sum of the two different numbers [suppose x and y] can be expressed as the sum of (n + 1 ). The theorem is used to solve complex problems. In addition, Pascal’s Triangle forms an important aspect of the theorem that can be used to determine the binomial coefficients.

### Properties of the Binomial Expansion

The Binomial Theorem consists of many properties. Some of them are mentioned below. They are as follows:

• The total number of terms in the binomial expansion of (x + a)n is (n + 1).
• The sum of indices of a and x in every term is n.
• It is considered a correct expansion when the terms are complex numbers.
• Terms that are equidistant from each other will have equal coefficients. They are termed binomial coefficients.
• General term in the expansion of (x + c)n is Tr + 1 = nCrx n – r ar .
• The values will first increase and then it will decrease when one will go ahead with the expansion.
• The coefficient of xr in the expansion of (1 + x)n is nCr.

### Middle Term in the Expansion of (1 + x)n

• In (a + b)n if n is even then the term expansion is odd. Hence, there will be only one middle term and it is (n + 2 / 2)th term.
• In (a + b)n if n is the odd number then term expansion will be even. Therefore, there will be two middle terms (n +1 / 2)th and (n + 3 / 2)th

### Greatest Coefficient

• If n is an even term then in (x + a)n the greatest coefficient will be nCn/2.
• If n is odd then in (x + a)n the greatest coefficient will be nCn – ½ or nCn + ½ where both are equal.

### Fun Facts about the Greatest Term

Do you want to learn methods to find the greatest term? If yes then you can easily find the greatest coefficient for each expansion by determining whether the resulting expansion is an integer or not. To find the answers accurately students should have a good hold over the definition of integers and must memorise the formulas to solve the questions.

### 1. Why should I refer to the CBSE Class 11 Chapter 8 Notes?

Students should refer to the Class 11 Chapter 8 notes by Extramarks as these will:

• Help the candidates to prepare well for the examination.
• Students will be able to understand the basic and advanced concepts.
• They can score good marks if they follow the notes properly.

### 2. Mention two applications of the Binomial Theorem.

The theorem is used in areas such as probability, and statistics and to find the remaining digits of a number.

### 3. Mention some important points from the Binomial theorem.

The most important points of the theorem are mentioned below. They are as follows:

• The total terms used in the expansion of (x + y)n are (n + 1).
• The total sum of x and y exponents is always n.
• nC0, nC1, nC2, … .., nCn are known as the binomial coefficients and can also be written as C0, C1, C2, ….., Cn.

The Binomial coefficients that are equidistant from the beginning and ending are always equal such as nC0 = nCn, nC1 = nCn-1 , nC2 = nCn-2 ,….. etc.