Question

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

• A. $1$
• B. $2$
• C. $\frac{1}{2}$
• D. $\frac{1}{n}$

Answer 1

The correct option is B $2$

We know that coefficient of $x^r$ in the expansion of ${(1+x)}^m$ is ${{}^{m}C}_r$
Thus $A={{}^{2n}C}_n;B={{}^{2n-1}C}_n$
We have
$\frac{A}{B}=\frac{{{}^{2n}C}_n}{{{}^{2n-1}C}_n}=\frac{\left(2n\right)!}{n!n!}\frac{(n!)(n-1)!}{(2n-1)!}=\frac{2n}{n}=2$