# If the sum of the coefficients in the expansion of ${\left( {x + y} \right)^n}$ is 4096, find the greatest coefficient in the expansion.

Last updated date: 15th Mar 2023

•

Total views: 305.4k

•

Views today: 2.85k

Answer

Verified

305.4k+ views

Hint: Sum of coefficients of ${\left( {x + y} \right)^n}$ is obtained when we put $x = y = 1$. And the greatest coefficient is the coefficient of the middle term(s) in its binomial expansion.

According to the question, the sum of coefficients in the expansion of ${\left( {x + y} \right)^n}$ is 4096.

We know that the sum of coefficients is the value of the expansion if we put all the variables equal to 1. Hence here we will put $x = y = 1$. So, we have:

$

\Rightarrow {\left( {1 + 1} \right)^n} = 4096, \\

\Rightarrow {2^n} = 4096, \\

\Rightarrow {2^n} = {2^{12}}, \\

\Rightarrow n = 12 \\

$

Since $n = 12$, the expansion is of ${\left( {x + y} \right)^{12}}$ and it will have a total of 13 terms.

We know that the greatest coefficient is the middle term. In this case, it will be of 7th term.

The general term for binomial expansion of ${\left( {x + y} \right)^{12}}$ is:

$ \Rightarrow {T_{r + 1}}{ = ^{12}}{C_r}{x^{12 - r}}.{y^r}$

For middle term (i.e. 7th term), we will put $r = 6$:

$ \Rightarrow {T_7}{ = ^{12}}{C_6}{x^6}.{y^6}$

Thus the coefficient of the middle term is $^{12}{C_6} = 924$

And hence the greatest coefficient in the expansion is 924.

Note:

In the expansion of ${\left( {x + y} \right)^n}$, coefficient of the middle term is $^n{C_{\dfrac{n}{2}}}$ if $n$ is even.

But if $n$ is odd, there will be two middle terms having coefficients $^n{C_{\dfrac{{\left( {n - 1} \right)}}{2}}}$ and $^n{C_{\dfrac{{\left( {n + 1} \right)}}{2}}}$. The value of the coefficients will be the same though.

According to the question, the sum of coefficients in the expansion of ${\left( {x + y} \right)^n}$ is 4096.

We know that the sum of coefficients is the value of the expansion if we put all the variables equal to 1. Hence here we will put $x = y = 1$. So, we have:

$

\Rightarrow {\left( {1 + 1} \right)^n} = 4096, \\

\Rightarrow {2^n} = 4096, \\

\Rightarrow {2^n} = {2^{12}}, \\

\Rightarrow n = 12 \\

$

Since $n = 12$, the expansion is of ${\left( {x + y} \right)^{12}}$ and it will have a total of 13 terms.

We know that the greatest coefficient is the middle term. In this case, it will be of 7th term.

The general term for binomial expansion of ${\left( {x + y} \right)^{12}}$ is:

$ \Rightarrow {T_{r + 1}}{ = ^{12}}{C_r}{x^{12 - r}}.{y^r}$

For middle term (i.e. 7th term), we will put $r = 6$:

$ \Rightarrow {T_7}{ = ^{12}}{C_6}{x^6}.{y^6}$

Thus the coefficient of the middle term is $^{12}{C_6} = 924$

And hence the greatest coefficient in the expansion is 924.

Note:

In the expansion of ${\left( {x + y} \right)^n}$, coefficient of the middle term is $^n{C_{\dfrac{n}{2}}}$ if $n$ is even.

But if $n$ is odd, there will be two middle terms having coefficients $^n{C_{\dfrac{{\left( {n - 1} \right)}}{2}}}$ and $^n{C_{\dfrac{{\left( {n + 1} \right)}}{2}}}$. The value of the coefficients will be the same though.

Recently Updated Pages

Calculate the entropy change involved in the conversion class 11 chemistry JEE_Main

The law formulated by Dr Nernst is A First law of thermodynamics class 11 chemistry JEE_Main

For the reaction at rm0rm0rmC and normal pressure A class 11 chemistry JEE_Main

An engine operating between rm15rm0rm0rmCand rm2rm5rm0rmC class 11 chemistry JEE_Main

For the reaction rm2Clg to rmCrmlrm2rmg the signs of class 11 chemistry JEE_Main

The enthalpy change for the transition of liquid water class 11 chemistry JEE_Main

Trending doubts

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Write a letter to the Principal of your school to plead class 10 english CBSE