Question

If $A$ and $B$ are the coeff of $x^n$ in the expansion of $(1+x{)}^{2n}$ and $(1+x{)}^{2n-1}$ respectively, then $\frac{A}{B}$ equals:

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

Answer 1

Answer: (B)

$2$

Solution:
Coefficient of $x^n$ in the expansion $(1+x{)}^{2n}$ i.e. $A={}^{2n}{C}_n$

and coefficient of $x^n$ in the expansion $(1+x{)}^{2n-1}$ i.e. $B={}^{2n-1}{C}_n$

Now, $\frac{A}{B}=$$\frac{{}^{2n}{C}_n}{{}^{2n-1}{C}_n}=\frac{2n!n!(2n-1-n)!}{n!(2n-n)!(2n-1)!}=\frac{2n(n-1)!}{n!}=\frac{2n}{n}=\frac{2}{1}$

Hence, B is the correct answer.