Question

The coeff of $x^n$ in the expansion of $(1+x{)}^{2n}$ and $(1+x{)}^{2n-1}$ are in the ratio:

• A. $1:2$
• B. $1:3$
• C. $3:1$
• D. $2:1$

Answer 1

Answer: (D)

$2:1$

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

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

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

Hence, D is the correct answer.