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

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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.
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