** Binomial Theorem **is a theorem that is used to find the expansion of algebraic identity (ax + by)

^{n}. We can easily find the expansion of (x + y)

^{2}, (x + y)

^{3,}and others but finding the expansion of (x + y)

^{21}is a tedious task and this task can easily be achieved using the Binomial Theorem or Binomial Expansion.

As the Binomial theorem is used to find the expansion of ** two terms** it is called the Binomial Theorem. The exponent value in the Binomial expansion can also be a negative value or fraction value.

Let’s learn about **the Binomial theorem, its expansion, formula, and others in detail in this article.**

Table of Content

- What is the Binomial Theorem?
- Binomial Expansion
- Binomial Theorem Formula
- Binomial Theorem Expansion Proof
- Properties of Binomial Theorem
- Pascal’s Triangle Binomial Expansion
- Binomial Expansion for Negative Exponent
- Important Terms of Binomial Theorem
- Applications of Binomial Theorem
- Binomial Theorem for Any Index
- Multinomial Theorem
- Binomial Distribution and Binomial Coefficients

**What is the Binomial Theorem?**

**What is the Binomial Theorem?**

The binomial theorem is used to solve binomial expressions simply. This theorem was first used somewhere around 400 BC by Euclid, a famous Greek mathematician.

It gives an expression to calculate the expansion of algebraic expression (a+b)^{n}. The terms in the expansion of the following expression are exponent terms and the constant term associated with each term is called the coefficient of terms.

### Binomial Theorem Statement

The Binomial theorem for the expansion of (a+b)^{n }is stated as,

(a + b)^{n}=^{n}C_{0 }a^{n}b^{0}+^{n}C_{1 }a^{n-1 }b^{1}+^{n}C_{2 }a^{n-2 }b^{2}+ …. +^{n}C_{r }a^{n-r }b^{r}+ …. +^{n}C_{n }a^{0}b^{n}

where n>0 and the ^{n}C_{k} is the binomial coefficient.

**Example: Find the expansion of (x + 5)**^{6}** using the binomial theorem.**

**Solution:**

We know that here,

(a + b)

^{n}=^{n}C_{0 }a^{n}b^{0}+^{n}C_{1 }a^{n-1 }b^{1}+^{n}C_{2 }a^{n-2 }b^{2}+ …. +^{n}C_{r }a^{n-r }b^{r}+ …. +^{n}C_{n }a^{0}b^{n}Thus, (x + 5)

^{6}=^{6}C_{0 }x^{6}5^{0}+^{6}C_{1}x^{6-1}5^{1}+^{6}C_{2}x^{6-2}5^{2}+^{6}C_{3}x^{6-3}5^{3}+^{6}C_{4}x^{6-4}5^{4}+^{6}C_{5}x^{6-5}5^{5}+^{6}C_{6}x^{6-6}5^{6}Â =

^{6}C_{0 }x^{6}+^{6}C_{1}x^{5}5 +^{6}C_{2}x^{4}5^{2}+^{6}C_{3}x^{3}5^{3}+^{6}C_{4}x^{2}5^{4}+^{6}C_{5}x^{1}5^{5}+^{6}C_{6}x^{0}5^{6}Â = x

^{6}+ 30x^{5}+ 375x^{4}+ 2500x^{3}+ 9375x^{2}+ 18750x + 15625

## Binomial Expansion

Binomial Theorem is used to expand the algebraic identity (x + y)^{n}. Hence it is also called the binomial expansion. The binomial expansion of (x + y)^{n} with the help of the binomial theorem is,

(x+y)n = nC0 xny0 + nC1 xn-1y1 + nC2 xn-2 y2 + … + nCn-1 x1yn-1 + nCnx0yn

Using this expansion we get each term in the expansion of (x+y)^{nl}

**Example: Find the value of (x+y)**^{2}** and (x+y)**^{3}** using Binomial expansion.**

**Solution:**

(x+y)

^{2}=^{2}C_{0}x^{2}y^{0}+^{2}C_{1}x^{2-1}y^{1}+^{2}C_{2}x^{2-2}y^{2}â‡’ (x+y)

^{2 }= x^{2}+ 2xy + y^{2}And (x+y)

^{3}=^{3}C_{0}x^{3}y^{0}+^{3}C_{1}x^{3-1}y^{1}+^{3}C_{2}x^{3-2}y^{2}+^{3}C_{3}x^{3-3}y^{3}â‡’ (x+y)

^{3 }= x^{3}+ 3x^{2}y + 3xy^{2}+ y^{3}â‡’ (x+y)

^{3 }= x^{3}+ 3xy(x+y) + y^{3}

## Binomial Theorem Formula

The binomial theorem formula gives the expansion of the algebraic identities in the form of a series. This formula is used to find the expansion up to n terms of the (a+b)^{n}. The binomial expansion of (a+b)^{n} can easily be represented with the summation formula.

The Binomial Theorem Formula for the expansion of (a+b)^{n} is,

(a+b)^{n}= âˆ‘^{n}_{r }^{n}C_{r}a^{n-r }b^{r}Where,

is a positive integer,nare real numbers, anda, b0 < r â‰¤ n

We can easily find the expansion of the various identities such as (x+y)^{7}, (x+9)^{11}, and others using the Binomial Theorem Formula. We can also find the expansion of (ax + by)^{n} using the Binomial Theorem Formula,

The expansion formula for (ax + by)^{n} is,

(ax + by)^{n}= âˆ‘^{n}_{r }^{n}C_{r}(ax)^{n-r }(by)^{r}Where

0< r â‰¤ n.

Also using the combination formula we know that,

^{n}C_{r}= n! / [r! (n – r)!]

## Binomial Theorem Expansion Proof

We can easily proof Binomial Theorem Expansion using the concept of Principle of Mathematical Induction

Let’s take x, a, n âˆˆ N, and then the Binomial Theorem says that,

**(x+y)**^{n}** = **^{n}**C**_{0}** x**^{n}**y**^{0}** + **^{n}**C**_{1}** x**^{n-1}**y**^{1}** + **^{n}**C**_{2}** x**^{n-2}** y**^{2}** + … + **^{n}**C**_{n-1}** x**^{1}**y**^{n-1}** + **^{n}**C**_{n}** x**^{0}**y**^{n}

**Proof:**

For n = 1, we get

(x +y)

^{1}= x +y which is true.For n = 2,

(x +y)

^{2}= (x + y) (x +y)

â‡’ (x +y)^{2}= x^{2}+ xy + xy + y^{2}Â Â Â Â Â Â Â Â Â Â (using distributive property of multiplication over addition)

â‡’ (x +y)^{2}= x^{2}+ 2xy + y^{2}which is also true.

Thus the theorem is true for n = 1 and n = 2.

Let’s take a positive integer k.

(x+y)

^{k}=^{k}C_{0}x^{k}y^{0}+^{k}C_{1}x^{k-1}y^{1}+^{k}C_{2}x^{k-2}y^{2}+ … +^{k}C_{k-1}x^{1}y^{k-1}+^{k}C_{k}x^{0}y^{k}Now consider the expansion for n = k + 1

(x + y)

^{k+1}= (x + y) (x + y)^{k}â‡’ (x + y)

^{k+1}= (x + y) (x^{k}+^{k}C_{1}x^{k-1}y^{1}+^{ k}C_{2}x^{k-2}y^{2}+ … +^{k}C_{r}x^{k-r}y^{r}+….+ y^{k})â‡’ (x + y)

^{k+1}= x^{k+1}+ (1 +^{k}C_{1})x^{k}y + (^{k}C_{1}+^{k}C_{2}) x^{k-1}y^{2}+ … + (^{k}C_{r-1}+^{k}C_{r}) x^{k-r+1}y^{r }+ … + (^{k}C_{k-1}+ 1) xy^{k}+ yk^{+1}â‡’ (x + y)

^{k+1}= x^{k+1}+^{k+1}C_{1}x^{k}y +^{k+1}C_{2}x^{k-1}y^{2}+ … +^{k+1}C_{r}x^{k-r+1}y^{r}+ … +^{k+1}C_{k}xy^{k}+ y^{k+1}Â Â[As we know,

^{n}C_{r}+^{n}C_{r-1}=^{n+1}C_{r}]This result is true for n = k+1.

Thus, by the concept of mathematical induction, the result is true for all positive integers ‘n’. Proved.

## Properties of Binomial Theorem

There are various properties related to the binomial theorem, some of those properties are as follows:

In the binomial Expansion of (x + y)**Number of Terms:**^{n}using the binomial theorem, there are n+1 Â terms and coefficients.- The first and last terms in the binomial expansion of (x + y)
^{n}are x^{n}and y^{n}, respectively. In the binomial expansion of (x + y)**General Term:**^{n}the general term can be represented as T(r + 1) and is given by T(r + 1) =^{n}C_{r}Ã— x^{(n-r)}Ã— y^{r}.The binomial coefficients in the expansion are arranged in Pascal’s triangle which is the pattern of number where each number is the sum of the two numbers above it.**Pascal’s Triangle:**- Â
When n is a non-negative integer, the expansion simplifies for specific values of n:**Specific Values:**- (a + b)
^{0}= 1 - (a + b)
^{1}= a + b - (a + b)
^{2}= a^{2}+ 2ab + b^{2} - (a + b)
^{3}= a^{3}+ 3a^{2}b + 3ab^{2}+ b^{3} - and so on…

- (a + b)
In the binomial expansion of (x + y)**Binomial Coefficients Symmetry:**^{n}, the r^{th}term from the end is the (n – r + 2)^{th}term from the beginning.- If n is even, then in the expansion of (x + y)
^{n}, the middle term is ((n/2) + 1). If n is odd, then the middle terms are ((n + 1)/2) and ((n + 3)/2) in the expansion of (x + y)^{n}.

## Pascal’s Triangle Binomial Expansion

The number associated with the terms of the binomial expansion is called the coefficient of the binomial expansion. These variables can easily be found using Pascal’s Triangle or by using combination formulas.

### Binomial Theorem Coefficients

After examining the coefficient of the various terms in the expansion of algebraic identities using the binomial theorem, we have observed a trend in the coefficient of the terms of the expansion. The trend is that the coefficient of the terms in the expansion of the binomial terms is directly similar to the rows of the Pascal triangle.

For example, the coefficient of each term in the expansion of the (x+y)^{4} is directly equivalent to the terms in the 4^{th} row of the Pascal Triangle.

Now, let’s learn about Pascal Triangle in detail.

### Pascal Triangle

** Pascal Triangle** is an alternative method of the calculation of coefficients that come in binomial expansions, using a diagram rather than algebraic methods.

In the diagram shown below, it is noticed that each number in the triangle is the sum of the two directly above it. This pattern continues indefinitely to obtain coefficients of any index of the binomial expression.

When we observe the pattern of the coefficients of the expansion (a + b)^{n}, ** Pascal’s triangle** for the pattern of the coefficients of the expansion (a + b)

^{7 }is shown in the figure below:

**Learn More: ****Pascalâ€™s Triangle**

Thus, from the above diagram, the expansion of small powers of n (e.g. n=0, 1, 2, 3, 4, 5, 6, 7) can be calculated as:

(a + b)^{0}= 1(a + b)^{1 }= a + b(a + b)^{2}= a^{2 }+ 2ab +b^{2}(a + b)^{3}= a^{3}+ 3a^{2}b + 3ab^{2}+ b^{3}- (a + b)
^{4}= a^{4}+ 4a^{3}b + 6a^{2}b^{2}+ 4ab^{3}+ b^{4}- (a + b)
^{5}= a^{5}+ 5a^{4}b + 10a^{3}b^{2}+ 10a^{2}b^{3}+ 5ab^{4}+ b^{5}- (a + b)
^{6}= a^{6}+ 6a^{5}b + 15a^{4}b^{2}+ 20a^{3}b^{3}+ 15a^{2}b^{4}+ 6ab^{5}+ b^{6}- (a + b)
^{7}= a^{7}+ 7a^{6}b + 21a^{5}b^{2}+ 35a^{4}b^{3}+ 35a^{3}b^{4}+ 21a^{2}b^{5}+ 7ab^{6}+ b^{7}

In this way, the expansion from (a + b)^{0} to (a + b)^{7} is obtained by the application of Pascal’s Triangle. But finding (a + b)^{15} is really a long process using Pascal’s triangle. So, here we use combinations formulas.

### Combinations

The combination formulas are also widely used to find coefficients of the various terms in the expansion of the binomial theorem.

The combination formula used for choosing ** r** objects out of

**total objects is widely used in the binomial expansion and it is defined as,**

**n**

^{n}C_{r}= n! / [r! (n – r)!]

Also, some common combination formulas used in the Binomial Expansion are,

^{n}C_{n}=^{n}C_{0}= 1^{n}C_{1}=^{n}C_{n-1}= n^{n}C_{r}=^{n}C_{r-1}

Some properties of combination which are widely used in simplifying binomial expansion are,

- C
_{1}+ C_{2}+ C_{3}+ C_{4}+ …….C_{n}= 2^{n} - C
_{0}+ C_{2}+ C_{4}+ …. = C_{1}+ C_{3}+ C_{5}+ ……. = 2^{n-1} - C
_{0}– C_{1 }+ C_{2}– C_{3 }+ C_{4}– C_{5 }+ …. = 0 - C
_{1}+ 2C_{2}+ 3C_{3}+ 4C_{4}+ …….nC_{n}= n2^{n-1} - C
_{1}– 2C_{2}+ 3C_{3}– 4C_{4}+ …….(-1)^{n}nC_{n}= 0 - C
_{1}^{2}+ C_{2}^{2}+ C_{3}^{2}+ C_{4}^{2}+ …….C_{n}^{2}= (2n)! / (n!)^{2}

**Learn more about, ****Permutations and Combinations**

## Binomial Expansion for Negative Exponent

The binomial theorem is also used for finding the expansion of the identities which have negative exponents. The coefficients terms in the negative expansion are similar in magnitude to the terms in the corresponding positive exponent.

Some of the simplified expansions of the negative exponents which are widely used are,

- (1 + x)
^{-1}= 1 – x + x^{2}– x^{3}+ x^{4}– x^{5}+ ……. - (1 – x)
^{-1}= 1 + x + x^{2}+ x^{3}+ x^{4}+ x^{5}+ ……. - (1 + x)
^{-2}= 1 – 2x + 3x^{2}– 4x^{3}+ …….. - (1 – x)
^{-2}= 1 + 2x + 3x^{2}+ 4x^{3}+ ……..

These expansions are easily proved using binomial expansion and replacing (+) with (-)

## Important Terms of Binomial Theorem

Various terms related to Binomial Expansion that is used widely include,

### General Term of Expansion

The general term of the binomial expansion signifies a term that produces all the terms of the binomial expansion by simply replacing the value of one component of the term.

For example in the binomial expansion of (x+y)^{n }the general term is,

T_{r+1}=^{n}C_{r}x^{n-r}y^{r}Where,

0 â‰¤ r â‰¤n

Substituting the values of r in the above term we get all the terms of the expansion. We observe that r takes n+1 values which is also true as there are n+1 terms in the expansion of (x+y)^{n}

### Middle Term of Expansion

We know that the total number of terms in the expansion of (x + y)^{n} is n + 1. And so the middle term in the binomial expansion of the (x+y)^{n} depends on the value of n. The value of n can be either even or odd which decides the number/numbers and values of the middle term.

- If n is even then we have odd (n+1) terms in the expansion of (x+y)
^{n}. Thus, there is one middle term, in this case, and the middle term is (n/2 + 1)^{th}term. - If n is odd then we have even (n+1) terms in the expansion of (x+y)
^{n}. Thus, there are two middle terms, in this case, and the middle terms are the (n/2)^{th}term and (n/2 + 1)^{th}term.

**Learn more about, ****General and Middle Terms**

### Identifying a Particular Term in Expansion

Any particular term can easily be identified in the expansion of (x+y)^{n}. We follow two steps to get a particular term in the expansion of (x+y)^{n}.

For example, if we have to find the term p^{th} term in the expansion of (x+y)^{n}.

** Step 1:** Find the general term in the expansion of (x+y)

^{n}. The general term is,

T_{r+1}=^{n}C_{r}x^{n-r}y^{r}

** Step 2: **Then substitute the value of ‘r’ for the value of the term which we have to find. In this case, r = p and then simplify to get the p

^{th}term.

**Example: Find the 6**^{th }**term in the expansion of (3x + 4)**^{8}

**Solution:**

The general term in the expansion of (3x + 4)

^{8}is,T

_{r+1}=^{8}C_{r}(3x)^{8-r}(4)^{r}For 6

^{th }term r = 5T

_{6 }= T_{5+1}=^{8}C_{5}(3x)^{8-5}(4)^{5}Simplifying the above term we get our answer,

T

_{6}=^{8}C_{5}(3x)^{8-5}(4)^{5}= (8.7.6/3.2.1)(27)(1024)x

^{3}= 1593648x

^{3}

### Term Independent of X in Expansion

The term independent of ‘x’ can easily be identified in the expansion of (x+y)^{n}. We follow two steps to get a particular term in the expansion of (x+y)^{n}.

For example, if we have to find the term independent of ‘x’ term in the expansion of (x+y)^{n}.

** Step 1:** Find the general term in the expansion of (x+y)

^{n}. the general term is,

T_{r+1}=^{n}C_{r}x^{n-r}y^{r}

** Step 2: **Then substitute the value of ‘r’ Â as r = n to get rid of the x term.

**Example: Find the**^{ }**term independent of x in the expansion of (2x + 1)**^{8}

**Solution:**

The general term in the expansion of (2x + 1)

^{8}is,T

_{r+1}=^{8}C_{r}(2x)^{8-r}(1)^{r}For the term independent of x

8-r = 0

r = 8

T

_{9 }= T_{8+1}=^{8}C_{8}(2x)^{8-8}(1)^{8}Simplifying the above term we get our answer,

T

_{9}=^{8}C_{8}(2x)^{8-8}(1)^{8}= (1)(1)(1)

= 1

### Numerically Greatest Term of Expansion

We can easily find the numerically greatest term in the expansion of (1+x)^{n}. The formula to find the greatest term in the expansion of (1+x)^{n} is,

[(n+1)|x|]/(1+|x|)

While using this formula we have to make sure that the expansion is in (1+x)^{n} form only, and |x| gives only the numerical value.

## Applications of Binomial Theorem

The Binomial Theorem has various applications it is used for a variety of the purposes such as,

- Finding Remainder in the division of very large numbers.
- Finding Last Digits of a Number
- Checking the Divisibility

### Finding Remainder using Binomial Theorem

This can easily be understood with the help of the following example.

**Example: Find the remainder when 2**^{97}** is divided by 15**

**Solution:**

(2

^{97}/ 15) = [2(2^{4})^{24}/ 15)]Â= [2(15 + 1)

^{24}/ 15]As each term of the expansion of (1 + 15)

^{24}contains 15 except the first term which is only 1.Thus, Remainder when 2

^{97}is divided by 15 is 1.

### Finding the Last Digits of a Number

How to find the last digit of an expansion can be understood using the example,

**Example: Find the last digit of (7)**^{10}

**Solution:**

(7)

^{10}= (49)^{5}= (50-1)^{5}(50-1)

^{5}=^{5}C_{0}(50)^{5}âˆ’^{5}C_{1}(50)^{4}+ 5C_{2}(50)3 âˆ’^{5}C_{3}(50)^{2}+^{5}C_{4}(50) âˆ’^{5}C_{5}Â Â Â Â Â Â =

^{5}C_{0}(50)^{5}âˆ’^{5}C_{1}(50)^{4}+ 5C_{2}(50)3 âˆ’^{5}C_{3}(50)^{2}+ 5(50) âˆ’ 1Â Â Â Â Â Â =

^{5}C_{0}(50)^{5}âˆ’^{5}C_{1}(50)^{4}+ 5C_{2}(50)3 âˆ’^{5}C_{3}(50)^{2}+ Â 249A multiple of 50 + 249 = 50K + 249

Thus, the last digit is 9.

## Binomial Theorem for Any Index

The Binomial Theorem for any index, including non-integer and negative indices, generalizes the familiar binomial expansion that applies to positive integer exponents. This generalized form involves the use of binomial coefficients that are defined for any real number index using the concept of factorial functions extended to the gamma function for non-integer values.

## Multinomial Theorem

We know that the binomial theorem expansion of (x + a)^{n} is,

(x+a)^{n }=^{n}âˆ‘_{r}^{ n}C_{r}x^{n â€“ r}a^{r}Where

nâˆˆN

We can generalize this result to get the expansion of,

(x_{1} + x_{2 }+ â€¦ +x_{k})^{n} = âˆ‘_{(r1 + r2 + â€¦. + rk = n)} [n! / r_{1}!r_{2}!â€¦r_{k}!] x_{1}^{r1} x_{2}^{r2} â€¦x_{k}r^{k}

The general term in the above expansion is

Â [n! / r_{1}!r_{2}!â€¦r_{k}!] x_{1}^{r1} x_{2}^{r2} â€¦x_{k}r^{k}

The number of terms in the above expansion is equal to the number of non-negative integral solutions of the equation r_{1 }+ r_{2} + â€¦ + r_{k} = n

Each solution of this equation gives a term in the above multinomial expansion. The total number of solutions can be given by ^{n + k â€“ 1}C_{k âˆ’1}.

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## Binomial Distribution and Binomial Coefficients

### Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent and identical Bernoulli trials (experiments with two possible outcomes: success or failure). It’s characterized by two parameters: nn (the number of trials) and pp (the probability of success in each trial).

### Hypergeometric Distribution

The hypergeometric distribution models the probability of obtaining a specific number of successes in a sample of a fixed size drawn without replacement from a finite population containing a known number of successes and failures.

### Also Check:

### Binomial Theorem Class 11 NCERT Solutions

### Binomial Theorem Class 11 Notes

## Examples on Binomial Theorem

**Example 1: Expand the binomial expression (2x + 3y)**^{2}**.**

**Example 1: Expand the binomial expression (2x + 3y)**

^{2}

**.**

**Solution:**

(2x + 3y)

^{2}^{Â }(2x)=^{2}+ 2(2x)(3y) + (3y)^{2}Â Â Â Â Â Â Â Â Â = 4x

^{2}+ 12xy + 9y^{2}

**Example 2: Expand the following (1 – x + x**^{2}**)**^{4}

**Solution:**

Put 1 – x = y

Then,

(1 – x + x

^{2})^{4}= (y + x^{2})^{4}=

^{4}C_{0}y^{4}(x^{2})^{0 }+^{4}C_{1}y^{3}(x^{2})^{1 }+^{4}C_{2}y^{2}(x^{2})^{2 }+^{4}C_{3}y(x^{2})^{3 }+^{4}C_{4}(x^{2})^{4}= y

^{4 }+ 4y^{3}x^{2 }+ 6y^{2}x^{4 }+ 4yx^{6 }+ x^{8}= (1 – x)

^{4}+ 4(1 – x)^{3}x^{2}+ 6(1 – x)^{2}x^{4}+ 4(1 – x)x^{6}+ x^{8}= 1 – 4x + 10x

^{2}– 16x^{3}+ 19x^{4}– 16x^{5}+ 10x^{6}– 4x^{7}+ x^{8}

**Example 3: Find the 4th term from the end in the expansion of ((x**^{3}**/2) – (2/x**^{2}**))**^{8}**.**

**Solution:**

Since r

^{th}term from the end in the expansion of (a + b)^{n }is(n – r + 2)

^{th}term from the beginning.Therefore, 4

^{th}term from the end is 8 – 4 + 2,i.e., 6th term from the beginning, which is given by

T

_{6}=^{8}C_{5}(x^{3}/2)^{3}(-2/x^{2})^{5}=

^{8}C_{3}(x^{9}/8)(-32/x^{10})= -224/x

**Example 4: Find the middle term (terms) in the expansion of ((p/x) + (x/p))**^{9}

**Solution:**

Since the power of binomial is odd. Therefore, we have two middle terms

which are 5th and 6th terms. These are given by

T

_{5 Â }=^{9}C_{4}(p/x)^{5}(x/p)^{4}Â Â Â =

^{9}C_{4}(p/x)Â Â Â = 126(p/x)

T

_{6 Â }=^{9}C_{5}(p/x)^{4}(x/p)^{5}Â Â Â =

^{9}C_{5}(x/p)Â Â Â = 126(x/p)

**Example 5: Find the 3**^{th }**term in the expansion of (2x + 1)**^{4}

**Solution:**

The general term in the expansion of (2x + 1)

^{4}is,T

_{r+1}=^{4}C_{r}(2x)^{4-r}(1)^{r}For 3

^{th }term r = 2T

_{3 }= T_{2+1}=^{4}C_{2}(2x)^{4-2}(1)^{2}Simplifying the above term we get our answer,

T

_{3}=^{4}C_{2}(2x)^{4-2}(1)^{2}= (6)(4)x

^{2}= 24x

^{2}

**Example 6: Find the**^{ }**term independent of x in the expansion of (3x + 2)**^{8}

**Solution:**

The general term in the expansion of (2x + 1)

^{8}is,T

_{r+1}=^{8}C_{r}(3x)^{8-r}(2)^{r}For the term independent of x

8-r = 0

r = 8

T

_{9 }= T_{8+1}=^{8}C_{8}(3x)^{8-8}(2)^{8}Simplifying the above term we get our answer,

T

_{9}=^{8}C_{8}(3x)^{8-8}(2)^{8}= (1)(1)(256)

= 256

**Example 7: Find the binomial expansion of (x**^{3}** + 1)**^{3}**.**

**Solution:**

We know that,

(x + y)

^{n}=^{n}C_{0}x^{n}y^{0}+^{n}C_{1}x^{n-1}y^{1}+^{n}C_{2}x^{n-2}y^{2}+ … +^{n}C_{n-1}x^{1}y^{n-1}+^{n}C_{n}x^{0}y^{n}(x

^{3}+ 1)^{3}=^{3}C_{0}(x^{3})^{3}1^{0}+^{3}C_{1}Â (x^{3})^{2}1^{1}+^{3}C_{2}(x^{3})^{1}1^{2}+^{3}C_{3}(x^{3})^{0}1^{3}(x

^{3}+ 1)^{3}= (1)x^{9}+ (3)x^{6}+ (3)x^{3}+ (1)(x

^{3}+ 1)^{3}= x^{9}+ 3x^{6}+ 3x^{3}+ 1

## FAQs on Binomial Theorem

### What Is the Binomial Theorem?

The binomial theorem is the basic theorem which provides an expansion to algebraic identity such as (x+y)

^{n}

(x+y)^{n}=^{n}C_{0}x^{n}y^{0}+^{n}C_{1}x^{n-1}y^{1}+^{n}C_{2}x^{n-2}y^{2}+ … +^{n}C_{n-1}x^{1}y^{n-1}+^{n}C_{n}x^{0}y^{n}

### What Is the General Term in Binomial Theorem?

The general term is the binomial theorem expansion of (x+y)

^{n}is,

Â T_{r+1}=^{n}C_{r}x^{n-r}y^{r}where, 0â‰¤râ‰¤n

Substituting the values of r in the above term we get all the terms of the expansion.

### What Is the Constant Term in the Binomial Theorem?

The term in the binomial theorem expansion which is free from the variables is called the constant term.

### What is the Binomial Theorem formula?

The Binomial theorem is used to find the expansion of the algebraic terms of the form(x + y)

^{n}. The formula of the binomial expansion isÂ

(x + y)^{n}= Î£_{r}^{n}Â^{n}C_{r }x^{n â€“ r }Â· y^{r}

### What are the number of terms in the expansion of (x + a)^{n} + (x-a)^{n}?

The number of terms in the expansion of (x + a)

^{n}+ (x-a)^{n}is given with the help of the formula,

- If n is even
(n+2)/2Â- If n is odd
(n+1)/2