Question

State the whether given statement is true or false

Prove that the coefficient of xnxn

• A. True
• B. False

Answer 1

The correct option is A True

$(‘1+x{)}^{2n}$

general term = ${}^{2n}{C}_r{x}^r$

coefficient of $x^n={}^{2n}{C}_n$

$(1+x{)}^{n-1}$

General term = ${}^{(2n-1)}{C}_r{x}^r$

Coefficient of $x^n={}^{2n-1}{C}_n$

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

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

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

Coefficient of $x^n$ in $(1+x{)}^{2n}=(n(2n-1))$ times coefficient of $x^n$ in $(1+x{)}^{2n-1}$.