Question

The greatest co-efficient in the expansion of$ (1 + x)^{2n+2}$ is

• A. $\frac{2n!}{(n!)^2}$
• B. $\frac{(2n+2)!}{n!(n+1)!}$
• C. $\frac{(2n)!}{(n!(n+1))!}$
• D. $\frac{(2n+2)!}{((n+1)!)^2}$

Answer 1

Answer: (B)

Greatest co-eff. in $(1 + x)^{2n+2}$
$= \,{}^{2n+2}c_{n+1} = \frac{\left(2n+2\right)\,!}{\left(n+1\right)\,!\left(n+1\right)\,!}$
$= \frac{\left(2n+2\right)\,!}{\left[\left(n+1\right)\,!\right]^{2}}$
[$\because \,{}^{n}c_{r}$ is greatest if $r = \frac{n}{2}$ if $n$ is even]