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# Binomial Theorem

Using the direct multiplication we may expand , but as the power increases the expansion becomes lengthy and tedious to calculate.

A binomial expression(an expression containing two unlike terms, eg. ()) that has been raised to a very large power can be easily calculated with the help of the Binomial Theorem.Let us try to understand Binomal Theorem in detail in this article.

• Binomial Theorem
• General Term
• Some Important Points Related to Binomial Theorem
• Concept Video
• Pascalâ€™s Triangle
• Concept Video
• Some Important Expansions
• Middle Term
• Concept Video
• Practice Problems on Binomial Theorem
• FAQs on Binomial Theorem

## Binomial Theorem

Binomial Theorem is the method of expanding an expression that has been raised to any finite power. The binomial theorem has the expansion as follows:

,

Here,  and

( + 1) terms

We can observe that,

## General Term

term in the expansion of is known as a general term. Given by,

## Some Important Points Related to Binomial Theorem

• Number of terms in the expansion of
• Sum of powers in each term of the expansion is n.
•  are known as binomial coefficients.
• The binomial coefficients of the terms equidistant from the beginning and the end are equal  i.e.,
• Sum of all binomial coefficients  in the expansion of  is .

Proof :

Putting  in above equation, we get

• Sum of even term binomial coefficients = Sum of odd term binomial coefficients

Proof : Put  in the expansion formula i.e.

(sum of even or odd term binomial coefficients)

Sum of even or odd term binomial coefficients

• Coefficients and binomial coefficients are different.

Example:

Here, sum of coefficients

where as, sum of binomial coefficients

•
• If
• The  term from the end in the expansion of is  term from the beginning.

## Concept Video:

Properties of Binomial Coefficients | MATHS | JEE | Concept of the Day | GB Sir

## Pascalâ€™s Triangle

Pascalâ€™s Triangle is basically an arrangement of numbers that gives the coefficients in the expansion of any binomial expression. The numbers are so arranged that they reflect as a triangle. Firstly, 1 is placed at the top, and then we start putting the numbers below in a triangular pattern. The numbers which we get in each step are the addition of the above two numbers.

## Concept Video:

Pascal's Triangle | MATHS | JEE | Concept of the Day | GB Sir

For example, the pascalâ€™s triangle for the expansion of  is shown below:

 +                +      +                        +      +       +       +    +      +      +

## Some Standard Expansions

• ,

• ,

,

• ,

Example : Expand

Solution :

We know the binomial expansion as follows:

Here,

Now, }}

is the required expansion.

## Middle Term

In the expansion of  middle term can be found for two cases

Case ' is even

Total Number of terms  (odd)

Hence, middle term is

Case II :  is odd

Number of terms (even)

Hence, two middle terms are

## Concept Video:

General and Middle Terms of Binomial Expansion | MATHS | JEE | Concept of the Day | GB Sir

Numerically greatest term in the expansion of

• If  , where m is a positive integer then  will be the Numerically greatest terms (Both terms are equal)

• If , where  is a positive integer and then   is numerically the greatest term in the expansion of .

## Practice Problems on Binomial Theorem

Example : If the third term in the binomial expansion of  is  then find a possible value of x.

(a)            (b)                (c)               (d)

 Step 1 :   We know the general term in the binomial expansion of ( is So, third term, Put Step 2 :

Hence, the correct option is (b).

Example :

Find the term independent of  in the expansion of