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.

Table of Contents:

Binomial Theorem
General Term
Some Important Points Related to Binomial Theorem
Concept Video
Pascal’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

Pascal’s Triangle

Pascal’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 pascal’s triangle for the expansion of is shown below:

+ +

+ + +

+ + + +

Some Standard Expansions

Example : Expand $<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>3</mn><mi>x</mi></mfrac></mrow></mfenced><mn>4</mn></msup></math>$

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

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced open="[" close="]"><mfrac><mrow><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>+</mo><mn>1</mn></mrow><mn>2</mn></mfrac></mfenced><mrow><mi>t</mi><mi>h</mi></mrow></msup><mi>t</mi><mi>e</mi><mi>r</mi><mi>m</mi><mo> </mo><mo>=</mo><msup><mfenced open="[" close="]"><mfrac><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow><mn>2</mn></mfrac></mfenced><mrow><mi>t</mi><mi>h</mi></mrow></msup><mi>t</mi><mi>e</mi><mi>r</mi><mi>m</mi><mo>=</mo><msup><mfenced open="[" close="]"><mrow><mfrac><mi>n</mi><mn>2</mn></mfrac><mo>+</mo><mn>1</mn></mrow></mfenced><mrow><mi>t</mi><mi>h</mi></mrow></msup><mo> </mo><mi>t</mi><mi>e</mi><mi>r</mi><mi>m</mi><mspace linebreak="newline"/><mi>i</mi><mo>.</mo><mi>e</mi><mo>.</mo><mo>,</mo><mo> </mo><msub><mi>T</mi><mrow><mfrac><mi>n</mi><mn>2</mn></mfrac><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mmultiscripts><mi>C</mi><mfrac><mi>n</mi><mn>2</mn></mfrac><none/><mprescripts/><none/><mi>n</mi></mmultiscripts><msup><mi>x</mi><mfrac><mi>n</mi><mn>2</mn></mfrac></msup><msup><mi>y</mi><mfrac><mi>n</mi><mn>2</mn></mfrac></msup></math>$

Case II : is odd

Number of terms (even)

Hence, two middle terms are

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced><mfrac><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mn>2</mn></mfrac></mfenced><mrow><mi>t</mi><mi>h</mi></mrow></msup><mi>t</mi><mi>e</mi><mi>r</mi><mi>m</mi><mo> </mo><mo>&</mo><msup><mfenced><mrow><mfrac><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mn>2</mn></mfrac><mo>+</mo><mn>1</mn></mrow></mfenced><mrow><mi>t</mi><mi>h</mi></mrow></msup><mi>t</mi><mi>e</mi><mi>r</mi><mi>m</mi></math>$

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>.</mo><mi>e</mi><mo>.</mo><mo>,</mo><mo> </mo><msub><mi>T</mi><mfrac><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mn>2</mn></mfrac></msub><mo>=</mo><mmultiscripts><mi>C</mi><mfrac><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mn>2</mn></mfrac><none/><mprescripts/><none/><mi>n</mi></mmultiscripts><msup><mi>x</mi><mfrac><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mn>2</mn></mfrac></msup><msup><mi>y</mi><mfrac><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mn>2</mn></mfrac></msup><mspace linebreak="newline"/><mo>&</mo><mo> </mo><msub><mi>T</mi><mfrac><mrow><mi>n</mi><mo>+</mo><mn>3</mn></mrow><mn>2</mn></mfrac></msub><mo>=</mo><mmultiscripts><mi>C</mi><mfrac><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mn>2</mn></mfrac><none/><mprescripts/><none/><mi>n</mi></mmultiscripts><msup><mi>x</mi><mfrac><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mn>2</mn></mfrac></msup><msup><mi>y</mi><mfrac><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mn>2</mn></mfrac></msup></math>$

Concept Video:

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

Numerically greatest term in the expansion of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mi>n</mi></msup></math>

If $<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced open="|" close="|"><mi>x</mi></mfenced></mrow><mrow><mfenced open="|" close="|"><mi>x</mi></mfenced><mo>+</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mi>m</mi></math>$ , where m is a positive integer then will be the Numerically greatest terms (Both terms are equal)

If $<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced open="|" close="|"><mi>x</mi></mfenced></mrow><mrow><mfenced open="|" close="|"><mi>x</mi></mfenced><mo>+</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mi>m</mi><mo>+</mo><mi>f</mi><mo> </mo></math>$ , 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)

Answer:

Step 1 :

We know the general term in the binomial expansion of ( is

So, third term,

Put

Step 2 :

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mfenced><mn>2</mn></mfenced><mrow><mn>2</mn><msup><mi>t</mi><mn>2</mn></msup></mrow></msup><mo>=</mo><msup><mn>2</mn><mn>8</mn></msup><mspace linebreak="newline"/><mo>⇒</mo><mn>2</mn><msup><mi>t</mi><mn>2</mn></msup><mo>=</mo><mn>8</mn><mspace linebreak="newline"/><mo>⇒</mo><msup><mi>t</mi><mn>2</mn></msup><mo>=</mo><mn>4</mn><mspace linebreak="newline"/><mo>⇒</mo><mi>t</mi><mo>=</mo><mo>±</mo><mn>2</mn><mspace linebreak="newline"/><mo>∵</mo><msub><mi>log</mi><mn>2</mn></msub><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>t</mi><mspace linebreak="newline"/><mo>⇒</mo><msub><mi>log</mi><mn>2</mn></msub><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>±</mo><mn>2</mn><mspace linebreak="newline"/><mo>⇒</mo><mi>x</mi><mo>=</mo><msup><mn>2</mn><mrow><mo>±</mo><mn>2</mn></mrow></msup><mspace linebreak="newline"/><mo>⇒</mo><mi>x</mi><mo>=</mo><mn>4</mn><mo>,</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mspace linebreak="newline"/></math>$

Hence, the correct option is (b).

Example :

Find the term independent of in the expansion of $<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced open="[" close="]"><mrow><mfrac><mrow><mo>(</mo><mi>x</mi><mo> </mo><mo>+</mo><mo> </mo><mn>1</mn><mo>)</mo><mo> </mo></mrow><mfenced><mrow><msup><mi>x</mi><mfrac><mn>2</mn><mn>3</mn></mfrac></msup><mo>-</mo><msup><mi>x</mi><mfrac><mn>1</mn><mn>3</mn></mfrac></msup><mo>+</mo><mn>1</mn></mrow></mfenced></mfrac><mo>-</mo><mfrac><mrow><mo>(</mo><mi>x</mi><mo> </mo><mo>-</mo><mo> </mo><mn>1</mn><mo>)</mo><mo> </mo></mrow><mfenced><mrow><mi>x</mi><mo>-</mo><msup><mi>x</mi><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></msup></mrow></mfenced></mfrac></mrow></mfenced><mn>10</mn></msup><mo>,</mo><mo> </mo><mi>x</mi><mo> </mo><mo>≠</mo><mo> </mo><mn>1</mn></math>$

Answer:

Step 1 :

Simplifying the given expression:

$<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced open="[" close="]"><mrow><mfrac><mrow><mo>(</mo><mi>x</mi><mo> </mo><mo>+</mo><mo> </mo><mn>1</mn><mo>)</mo><mo> </mo></mrow><mfenced><mrow><msup><mi>x</mi><mfrac><mn>2</mn><mn>3</mn></mfrac></msup><mo>-</mo><msup><mi>x</mi><mfrac><mn>1</mn><mn>3</mn></mfrac></msup><mo>+</mo><mn>1</mn></mrow></mfenced></mfrac><mo>-</mo><mfrac><mrow><mo>(</mo><mi>x</mi><mo> </mo><mo>-</mo><mo> </mo><mn>1</mn><mo>)</mo><mo> </mo></mrow><mfenced><mrow><mi>x</mi><mo>-</mo><msup><mi>x</mi><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></msup></mrow></mfenced></mfrac></mrow></mfenced><mn>10</mn></msup><mspace linebreak="newline"/><mo>=</mo><msup><mfenced open="[" close="]"><mrow><mfrac><mrow><mo>(</mo><mi>x</mi><mo> </mo><mo>+</mo><mo> </mo><mn>1</mn><mo>)</mo><mfenced><mrow><msup><mi>x</mi><mfrac><mn>1</mn><mn>3</mn></mfrac></msup><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mrow><mfenced><mrow><msup><mi>x</mi><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>3</mn></mfrac></mstyle></msup><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><msup><mi>x</mi><mfrac><mn>2</mn><mn>3</mn></mfrac></msup><mo>-</mo><msup><mi>x</mi><mfrac><mn>1</mn><mn>3</mn></mfrac></msup><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfrac><mo>-</mo><mfrac><mrow><mfenced><mrow><msup><mi>x</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo> </mo><mo>-</mo><mo> </mo><mn>1</mn></mrow></mfenced><mfenced><mrow><msup><mi>x</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo> </mo><mo>+</mo><mo> </mo><mn>1</mn></mrow></mfenced></mrow><mrow><msup><mi>x</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mfenced><mrow><msup><mi>x</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo>-</mo><mn>1</mn></mrow></mfenced></mrow></mfrac></mrow></mfenced><mn>10</mn></msup><mspace linebreak="newline"/><mo>=</mo><msup><mfenced open="[" close="]"><mrow><mfrac><mrow><mo>(</mo><mi>x</mi><mo> </mo><mo>+</mo><mo> </mo><mn>1</mn><mo>)</mo><mfenced><mrow><msup><mi>x</mi><mfrac><mn>1</mn><mn>3</mn></mfrac></msup><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mfenced><mrow><msup><mfenced><msup><mi>x</mi><mfrac><mn>1</mn><mn>3</mn></mfrac></msup></mfenced><mn>3</mn></msup><mo>+</mo><msup><mn>1</mn><mn>3</mn></msup></mrow></mfenced></mfrac><mo>-</mo><mfrac><mfenced><mrow><msup><mi>x</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo> </mo><mo>+</mo><mo> </mo><mn>1</mn></mrow></mfenced><msup><mi>x</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></msup></mfrac></mrow></mfenced><mn>10</mn></msup><mspace linebreak="newline"/><mo>=</mo><msup><mfenced open="[" close="]"><mrow><mfrac><mrow><mo>(</mo><mi>x</mi><mo> </mo><mo>+</mo><mo> </mo><mn>1</mn><mo>)</mo><mfenced><mrow><msup><mi>x</mi><mfrac><mn>1</mn><mn>3</mn></mfrac></msup><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mfenced><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mfrac><mo>-</mo><mfrac><mfenced><mrow><msup><mi>x</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo> </mo><mo>+</mo><mo> </mo><mn>1</mn></mrow></mfenced><msup><mi>x</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></msup></mfrac></mrow></mfenced><mn>10</mn></msup><mspace linebreak="newline"/><mspace linebreak="newline"/><mo>=</mo><msup><mfenced open="[" close="]"><mrow><mfenced><mrow><msup><mi>x</mi><mfrac><mn>1</mn><mn>3</mn></mfrac></msup><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mfrac><mfenced><mrow><msup><mi>x</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo> </mo><mo>+</mo><mo> </mo><mn>1</mn></mrow></mfenced><msup><mi>x</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></msup></mfrac></mrow></mfenced><mn>10</mn></msup><mspace linebreak="newline"/><mo>=</mo><msup><mfenced open="[" close="]"><mrow><msup><mi>x</mi><mfrac><mn>1</mn><mn>3</mn></mfrac></msup><mo>-</mo><msup><mi>x</mi><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup></mrow></mfenced><mn>10</mn></msup></math>$

Step 2 :

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>l</mi><mo> </mo><mi>t</mi><mi>e</mi><mi>r</mi><mi>m</mi><mo>,</mo><mo> </mo><msub><mi>T</mi><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow></msub><mspace linebreak="newline"/><mo>=</mo><mmultiscripts><mi>C</mi><mi>r</mi><none/><mprescripts/><none/><mn>10</mn></mmultiscripts><msup><mfenced><msup><mi>x</mi><mfrac><mn>1</mn><mn>3</mn></mfrac></msup></mfenced><mrow><mn>10</mn><mo>-</mo><mi>r</mi></mrow></msup><msup><mfenced><msup><mi>x</mi><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup></mfenced><mi>r</mi></msup><mspace linebreak="newline"/><mo>=</mo><mmultiscripts><mi>C</mi><mi>r</mi><none/><mprescripts/><none/><mn>10</mn></mmultiscripts><msup><mfenced><mi>x</mi></mfenced><mfrac><mfenced><mrow><mn>10</mn><mo>-</mo><mi>r</mi></mrow></mfenced><mn>3</mn></mfrac></msup><msup><mfenced><mi>x</mi></mfenced><mrow><mo>-</mo><mfrac><mi>r</mi><mn>2</mn></mfrac></mrow></msup><mspace linebreak="newline"/><mo>=</mo><mmultiscripts><mi>C</mi><mi>r</mi><none/><mprescripts/><none/><mn>10</mn></mmultiscripts><msup><mfenced><mi>x</mi></mfenced><mfrac><mfenced><mrow><mn>20</mn><mo>-</mo><mn>5</mn><mi>r</mi></mrow></mfenced><mn>6</mn></mfrac></msup><mspace linebreak="newline"/><mi>F</mi><mi>o</mi><mi>r</mi><mo> </mo><mi>t</mi><mi>e</mi><mi>r</mi><mi>m</mi><mo> </mo><mi>i</mi><mi>n</mi><mi>d</mi><mi>e</mi><mi>p</mi><mi>e</mi><mi>n</mi><mi>d</mi><mi>e</mi><mi>n</mi><mi>t</mi><mo> </mo><mi>o</mi><mi>f</mi><mo> </mo><mi>x</mi><mo>,</mo><mo> </mo><mspace linebreak="newline"/><mi>p</mi><mi>o</mi><mi>w</mi><mi>e</mi><mi>r</mi><mo> </mo><mi>o</mi><mi>f</mi><mo> </mo><mi>x</mi><mo> </mo><mi>s</mi><mi>h</mi><mi>o</mi><mi>u</mi><mi>l</mi><mi>d</mi><mo> </mo><mi>b</mi><mi>e</mi><mo> </mo><mi>z</mi><mi>e</mi><mi>r</mi><mi>o</mi><mspace linebreak="newline"/><mo>⇒</mo><mfrac><mfenced><mrow><mn>20</mn><mo>-</mo><mn>5</mn><mi>r</mi></mrow></mfenced><mn>6</mn></mfrac><mo>=</mo><mn>0</mn><mspace linebreak="newline"/><mo>⇒</mo><mi>r</mi><mo>=</mo><mn>4</mn><mspace linebreak="newline"/><mo>∴</mo><mi>T</mi><mi>h</mi><mi>e</mi><mo> </mo><mi>r</mi><mi>e</mi><mi>q</mi><mi>u</mi><mi>i</mi><mi>r</mi><mi>e</mi><mi>d</mi><mo> </mo><mi>t</mi><mi>e</mi><mi>r</mi><mi>m</mi><mo> </mo><mi>i</mi><mi>s</mi><mo> </mo><msub><mi>T</mi><mn>5</mn></msub><mo>=</mo><mmultiscripts><mi>C</mi><mn>4</mn><none/><mprescripts/><none/><mn>10</mn></mmultiscripts><mspace linebreak="newline"/><mo>=</mo><mfrac><mrow><mn>10</mn><mo>×</mo><mn>9</mn><mo>×</mo><mn>8</mn><mo>×</mo><mn>7</mn></mrow><mrow><mn>4</mn><mo>×</mo><mn>3</mn><mo>×</mo><mn>2</mn><mo>×</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mn>210</mn><mspace linebreak="newline"/></math>$

Example : If where and , then which of the following is/are true?

(a) is even (b) (c) is odd (d)

Answer:

Given,

Let

By adding equation (i) and (ii), we get,

Also, given that,

So,

Using

In the above expression . So, we have,

As we can see that in the above expression, side is an even natural number.

Hence,= should also be an even natural number.

As =.So, =

Also, =, i.e., odd (So that L.H.S is even)

=and=

As we can see that, =

or =

Hence, options (c) and (d) are the correct answers.

Example : If $<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>a</mi><mi>n</mi></msub><mo> </mo><mo>=</mo><mo> </mo><munderover><mo>∑</mo><mrow><mi>r</mi><mo> </mo><mo>=</mo><mn>0</mn></mrow><mi>n</mi></munderover><mfrac><mn>1</mn><mrow><msup><maction actiontype="argument"><mrow/></maction><mi>n</mi></msup><msub><mi>C</mi><mi>r</mi></msub></mrow></mfrac></math>$ , then find the value of $<math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∑</mo><mrow><mi>r</mi><mo>=</mo><mo> </mo><mn>0</mn></mrow><mi>n</mi></munderover><mfrac><mi>r</mi><mrow><msup><maction actiontype="argument"><mrow/></maction><mi>n</mi></msup><msub><mi>C</mi><mi>r</mi></msub></mrow></mfrac></math>$ .

a) b) c) d)

Answer:

Given, $<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>a</mi><mi>n</mi></msub><mo> </mo><mo>=</mo><mo> </mo><munderover><mo>∑</mo><mrow><mi>r</mi><mo> </mo><mo>=</mo><mn>0</mn></mrow><mi>n</mi></munderover><mfrac><mn>1</mn><mrow><msup><maction actiontype="argument"><mrow/></maction><mi>n</mi></msup><msub><mi>C</mi><mi>r</mi></msub></mrow></mfrac></math>$

Let $<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>b</mi><mi>n</mi></msub><mo> </mo><mo>=</mo><munderover><mo>∑</mo><mrow><mi>r</mi><mo>=</mo><mo> </mo><mn>0</mn></mrow><mi>n</mi></munderover><mfrac><mi>r</mi><mrow><msup><maction actiontype="argument"><mrow/></maction><mi>n</mi></msup><msub><mi>C</mi><mi>r</mi></msub></mrow></mfrac></math>$

$<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>b</mi><mi>n</mi></msub><mo> </mo><mo>=</mo><munderover><mo>∑</mo><mrow><mi>r</mi><mo>=</mo><mo> </mo><mn>0</mn></mrow><mi>n</mi></munderover><mfrac><mi>r</mi><mrow><msup><maction actiontype="argument"><mrow/></maction><mi>n</mi></msup><msub><mi>C</mi><mi>r</mi></msub></mrow></mfrac><mo>=</mo><munderover><mo>∑</mo><mrow><mi>r</mi><mo>=</mo><mo> </mo><mn>0</mn></mrow><mi>n</mi></munderover><mfrac><mrow><mi>n</mi><mo> </mo><mo>-</mo><mfenced><mrow><mi>n</mi><mo>-</mo><mi>r</mi></mrow></mfenced></mrow><mrow><msup><maction actiontype="argument"><mrow/></maction><mi>n</mi></msup><msub><mi>C</mi><mi>r</mi></msub></mrow></mfrac><mo> </mo><mo>=</mo><mo> </mo><munderover><mo>∑</mo><mrow><mi>r</mi><mo>=</mo><mo> </mo><mn>0</mn></mrow><mi>n</mi></munderover><mfrac><mi>n</mi><mrow><msup><maction actiontype="argument"><mrow/></maction><mi>n</mi></msup><msub><mi>C</mi><mi>r</mi></msub></mrow></mfrac><mo> </mo><mo>-</mo><mo> </mo><munderover><mo>∑</mo><mrow><mi>r</mi><mo>=</mo><mo> </mo><mn>0</mn></mrow><mi>n</mi></munderover><mfrac><mrow><mi>n</mi><mo>-</mo><mi>r</mi></mrow><mrow><msup><maction actiontype="argument"><mrow/></maction><mi>n</mi></msup><msub><mi>C</mi><mi>r</mi></msub></mrow></mfrac></math>$

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msub><mi>b</mi><mi>n</mi></msub><mo> </mo><mo>=</mo><mi>n</mi><mo> </mo><munderover><mo>∑</mo><mrow><mi>r</mi><mo>=</mo><mo> </mo><mn>0</mn></mrow><mi>n</mi></munderover><mfrac><mn>1</mn><mrow><msup><maction actiontype="argument"><mrow/></maction><mi>n</mi></msup><msub><mi>C</mi><mi>r</mi></msub></mrow></mfrac><mo> </mo><mo>-</mo><mo> </mo><munderover><mo>∑</mo><mrow><mi>r</mi><mo>=</mo><mo> </mo><mn>0</mn></mrow><mi>n</mi></munderover><mfrac><mrow><mi>n</mi><mo>-</mo><mi>r</mi></mrow><mrow><msup><maction actiontype="argument"><mrow/></maction><mi>n</mi></msup><msub><mi>C</mi><mrow><mi>n</mi><mo>-</mo><mi>r</mi></mrow></msub></mrow></mfrac><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mo>∵</mo><msup><mo> </mo><mi>n</mi></msup><msub><mi>C</mi><mi>r</mi></msub><mo> </mo><mo>=</mo><msup><mo> </mo><mi>n</mi></msup><msub><mi>C</mi><mrow><mi>n</mi><mo>-</mo><mi>r</mi></mrow></msub></mrow></mfenced><mspace linebreak="newline"/><mo>⇒</mo><mo> </mo><msub><mi>b</mi><mi>n</mi></msub><mo> </mo><mo>=</mo><mo> </mo><mi>n</mi><msub><mi>a</mi><mrow><mi>n</mi><mo> </mo></mrow></msub><mo> </mo><mo>-</mo><mo> </mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mo> </mo><mn>0</mn></mrow><mi>n</mi></munderover><mfrac><mi>k</mi><mrow><msup><maction actiontype="argument"><mrow/></maction><mi>n</mi></msup><msub><mi>C</mi><mi>k</mi></msub></mrow></mfrac><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo>⇒</mo><mo> </mo><msub><mi>b</mi><mi>n</mi></msub><mo> </mo><mo>=</mo><mo> </mo><mi>n</mi><msub><mi>a</mi><mrow><mi>n</mi><mo> </mo></mrow></msub><mo>-</mo><msub><mi>b</mi><mi>n</mi></msub><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mo>∵</mo><mo> </mo><mi>k</mi><mo> </mo><mo>=</mo><mo> </mo><mi>n</mi><mo>-</mo><mi>r</mi><mo> </mo></mrow></mfenced><mo> </mo><mo> </mo></math>$

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><msub><mi>b</mi><mi>n</mi></msub><mo> </mo><mo>=</mo><mfrac><mi>n</mi><mn>2</mn></mfrac><msub><mi>a</mi><mi>n</mi></msub></math>$

Hence, option (b) is the correct answer.

Example : Find the value of $<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mrow><mn>1</mn><mo>!</mo><mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>!</mo></mrow></mfrac><mo> </mo><mo>+</mo><mfrac><mn>1</mn><mrow><mn>3</mn><mo>!</mo><mfenced><mrow><mi>n</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>!</mo></mrow></mfrac><mo>+</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>+</mo><mfrac><mn>1</mn><mrow><mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>!</mo><mn>1</mn><mo>!</mo></mrow></mfrac></math>$ , where is an even positive integer.

Answer:

Let $<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mrow><mn>1</mn><mo>!</mo><mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>!</mo></mrow></mfrac><mo> </mo><mo>+</mo><mfrac><mn>1</mn><mrow><mn>3</mn><mo>!</mo><mfenced><mrow><mi>n</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>!</mo></mrow></mfrac><mo>+</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>+</mo><mfrac><mn>1</mn><mrow><mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>!</mo><mn>1</mn><mo>!</mo></mrow></mfrac></math>$

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">S</mi><mo> </mo><mo>=</mo><mfrac><mn>1</mn><mrow><mi>n</mi><mo>!</mo></mrow></mfrac><mo> </mo><mfenced open="[" close="]"><mrow><mfrac><mrow><mi>n</mi><mo>!</mo></mrow><mrow><mn>1</mn><mo>!</mo><mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>!</mo></mrow></mfrac><mo> </mo><mo>+</mo><mfrac><mrow><mi>n</mi><mo>!</mo></mrow><mrow><mn>3</mn><mo>!</mo><mfenced><mrow><mi>n</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>!</mo></mrow></mfrac><mo>+</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>+</mo><mfrac><mrow><mi>n</mi><mo>!</mo></mrow><mrow><mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>!</mo><mn>1</mn><mo>!</mo></mrow></mfrac></mrow></mfenced><mspace linebreak="newline"/><mo>⇒</mo><mo> </mo><mi mathvariant="normal">S</mi><mo> </mo><mo>=</mo><mo> </mo><mfrac><mn>1</mn><mrow><mi>n</mi><mo mathvariant="italic">!</mo></mrow></mfrac><mfenced open="[" close="]"><mrow><msup><maction actiontype="argument"><mrow/></maction><mi>n</mi></msup><msub><mi>C</mi><mn>1</mn></msub><mo mathvariant="italic"> </mo><msup><mo mathvariant="italic">+</mo><mi>n</mi></msup><msub><mi>C</mi><mn mathvariant="italic">3</mn></msub><mo mathvariant="italic"> </mo><mo mathvariant="italic">+</mo><mo mathvariant="italic">.</mo><mo mathvariant="italic">.</mo><mo mathvariant="italic">.</mo><mo mathvariant="italic">.</mo><mo mathvariant="italic">.</mo><mo mathvariant="italic">.</mo><mo mathvariant="italic">.</mo><mo mathvariant="italic">+</mo><msup><maction actiontype="argument"><mrow/></maction><mi>n</mi></msup><msub><mi>C</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow></mfenced></math>$

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi mathvariant="normal">S</mi><mo> </mo><mo>=</mo><mo> </mo><mfrac><msup><mn>2</mn><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mrow><mi>n</mi><mo mathvariant="italic">!</mo></mrow></mfrac><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><msup><mo>∵</mo><mi>n</mi></msup><msub><mi>C</mi><mn>1</mn></msub><mo mathvariant="italic"> </mo><msup><mo mathvariant="italic">+</mo><mi>n</mi></msup><msub><mi>C</mi><mn mathvariant="italic">3</mn></msub><mo mathvariant="italic"> </mo><mo mathvariant="italic">+</mo><mo mathvariant="italic">.</mo><mo mathvariant="italic">.</mo><mo mathvariant="italic">.</mo><mo mathvariant="italic">.</mo><mo mathvariant="italic">.</mo><mo mathvariant="italic">.</mo><mo mathvariant="italic">.</mo><mo mathvariant="italic">+</mo><msup><maction actiontype="argument"><mrow/></maction><mi>n</mi></msup><msub><mi>C</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo> </mo><mo>=</mo><mo> </mo><msup><mn>2</mn><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced></math>$

FAQs on Binomial Theorem

Question 1.What is the method to prove the Binomial Theorem formula?

Answer: We can apply Principle of Mathematical Induction to prove the Binomial Theorem formula.

Question 2.What is the use of Binomial expansion formula?

Answer: Binomial expansion is used to calculate the high powers of a Binomial expression without actually using the method of direct multiplication.

Question 3.What is the total number of terms in .

Answer: The total number of terms is given by .

Question 4.Is Binomial theorem for fractional index also defined?

Answer: Yes, there is a separate binomial theorem for fractional index (power)