If the sum of coefficients in the expansion of ${\left(a+b\right)}^n$  is 4096, then the greatest coefficient in the expansion is

Given expansion ${\left(a+b\right)}^n$  is,

 sum of coefficients in ${\left(a+b\right)}^n=4096$

${}^n{C}_0{+}^n{C}_1{+}^n{C}_2{+}^n{C}_3+\mathrm{.......}{+}^n{C}_n=2^n$

$\Rightarrow 2^n=4096$  $\Rightarrow 2^n=2^{12}$  $\Rightarrow n=12$

Hence, the greatest coefficient in ${\left(a+b\right)}^n={}^{12}{C}_6=924$

[$\because {}^n{C}_r$ is greatest for $r=\frac{n}{2}$  when $n$  is even]