Question

The sum of the coefficient of the in the expansion of ${\left( {x + y} \right)^n}$ is 4096.The greatest coefficient in the expansion is
a.1024
b.924
c.824
d.724

Answer 1

We know that the sum of the coefficient of the expansion can be obtained by replacing the variable by one. By substituting we get the value of n . The greatest coefficient is the coefficient of the middle term and its given by ${}^n{C_{\dfrac{n}{2}}}$ if n is even and ${}^n{C_{\dfrac{{n + 1}}{2}}}or{}^n{C_{\dfrac{{n - 1}}{2}}}$ if n is odd and with that we can find the greatest coefficient.

Complete step-by-step answer:
We are given that the sum of the coefficient of the expansion ${\left( {x + y} \right)^n}$ is 4096
We know that the sum of the coefficient of the expansion can be obtained by replacing the variable by one
Therefore now lets substitute x = 1 and y = 1
$\Rightarrow {\left( {1 + 1} \right)^n} = 4096 \\\ \Rightarrow {2^n} = 4096 \\\$
Now let's write 4096 in terms of 2
So 4096 = ${2^{12}}$
$\Rightarrow {2^n} = {2^9}$
From this we get that n = 12
And now the greatest coefficient is the coefficient of the middle term
And the coefficient of the middle term is given by ${}^n{C_{\dfrac{n}{2}}}$ if n is even and ${}^n{C_{\dfrac{{n + 1}}{2}}}or{}^n{C_{\dfrac{{n - 1}}{2}}}$if n is odd
Here our n = 12 is even
So our coefficient of middle term is ${}^n{C_{\dfrac{n}{2}}}$
$\Rightarrow {}^n{C_{\dfrac{n}{2}}} = {}^{12}{C_{\dfrac{{12}}{2}}} = {}^{12}{C_6}$
$\Rightarrow {}^{12}{C_6} = \dfrac{{12*11*10*9*8*7}}{{1*2*3*4*5*6}} \\\ \Rightarrow {}^{12}{C_6} = 11*2*3*2*7 = 22*42 = 924 \\\$
Therefore the greatest coefficient is 924
The correct option is b.

Note: Points to remember in a binomial expansion
1.There are n+1 terms in the expansion of ${(x + y)^n}$ .
2.The degree of each term is n.
3.The powers on x begin with n and decrease to 0.
4.The powers on y begin with 0 and increase to n.
5.The coefficients are symmetric.