Question

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

Answer 1

Sum of all the coefficient of $(x+y{)}^n$ is $4096$

Sum $=2^n=4096$

When $x=1$ and $y=1$

So $n=12$.

[Note: Greatest Coefficient of $(1+x{)}^n$ is the Coefficient of Middle

Term ${}^n{C}_{\frac{n}{2}}$, if $n$ is even and if $n$ is odd then ${}^n{C}_{\frac{(n+1)}{2}}$ or ${}^n{C}_{\frac{(n-1)}{2}}]$

So, the greatest coefficient is ${}^{12}{C}_6=924$.