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]