Question

If A and B are the coefficients of $x^n$ in the expansions of $(1+x{)}^{2n}$ and $(1+x{)}^{2n-1}$ respectively, then

• A. $A=B$
• B. $A=2B$
• C. $2A=B$
• D. none of these

Answer 1

The correct option is B $A=2B$

$\frac{\text{Coefficient of}{x}^n\text{in expansion of}(1+x{)}^{2n}}{\text{Coefficient of}{x}^n\text{in expansion of}(1+x{)}^{2n-1}}$

$=\frac{{}^{2n}{C}_n}{{}^{(2n-1)}{C}_n}=\frac{\left(2n\right)!}{n!n!}\times \frac{(n-1)!n!}{(2n-1)!}$

$=\frac{\left(2n\right)(2n-1)!(n-1)!}{n(n-1)!(2n-1)!}=\frac{2n}{n}=2:1$

$\Rightarrow \frac{A}{B}=\frac{2}{1}\Rightarrow A=2B$.