Question

The greatest binomial coefficient in the expansion of ${(a+b)}^n$ is ${}^k{C}_m$ given that the sum of all the coefficients is equal to $4096$. Find $k-m$ ?

Answer 1

Let $a=b=1$
Hence sum of coefficients
$=2^n=4096$
Hence
$2^n=2^{12}$
$\Rightarrow n=12$
The greatest coefficient will be the coefficient of the middle term
$={}^{12}{C}_{\frac{12}{2}}$
$={}^{12}{C}_6$.